Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Percentage Accurate: 98.4% → 98.4%

Time: 18.7s

Alternatives: 26

Speedup: 1.0×

• 46.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * exp((((y * log(z)) + ((t - 1.0d0) * log(a))) - b))) / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.exp((((y * Math.log(z)) + ((t - 1.0) * Math.log(a))) - b))) / y;
}

Python

def code(x, y, z, t, a, b):
	return (x * math.exp((((y * math.log(z)) + ((t - 1.0) * math.log(a))) - b))) / y

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Initial Program: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * exp((((y * log(z)) + ((t - 1.0d0) * log(a))) - b))) / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.exp((((y * Math.log(z)) + ((t - 1.0) * Math.log(a))) - b))) / y;
}

Python

def code(x, y, z, t, a, b):
	return (x * math.exp((((y * math.log(z)) + ((t - 1.0) * math.log(a))) - b))) / y

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t - 1.0) * log(a))) - b))) / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Alternative 1: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (* x (exp (- (+ (* y (log z)) (* (log a) (+ t -1.0))) b))) y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x * exp((((y * log(z)) + (log(a) * (t + -1.0))) - b))) / y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (x * exp((((y * log(z)) + (log(a) * (t + (-1.0d0)))) - b))) / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x * Math.exp((((y * Math.log(z)) + (Math.log(a) * (t + -1.0))) - b))) / y;
}

Python

def code(x, y, z, t, a, b):
	return (x * math.exp((((y * math.log(z)) + (math.log(a) * (t + -1.0))) - b))) / y

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(log(a) * Float64(t + -1.0))) - b))) / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x * exp((((y * log(z)) + (log(a) * (t + -1.0))) - b))) / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x * N[Exp[N[(N[(N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Derivation

1. Initial program 99.1%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
2. Add Preprocessing

3. Final simplification 99.1%

   \[\leadsto \frac{x \cdot e^{\left(y \cdot \log z + \log a \cdot \left(t + -1\right)\right) - b}}{y} \]
4. Add Preprocessing

Alternative 2: 64.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log a \cdot \left(t + -1\right)\\ t_2 := \mathsf{fma}\left(x, \frac{{a}^{t}}{y}, 0\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+29}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -340:\\ \;\;\;\;\frac{x \cdot \frac{\frac{a - a \cdot b}{a}}{a}}{y}\\ \mathbf{elif}\;t\_1 \leq -42:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (log a) (+ t -1.0))) (t_2 (fma x (/ (pow a t) y) 0.0)))
   (if (<= t_1 -1e+29)
     t_2
     (if (<= t_1 -340.0)
       (/ (* x (/ (/ (- a (* a b)) a) a)) y)
       (if (<= t_1 -42.0)
         (/ x (* y (exp b)))
         (if (<= t_1 1000.0) (* x (/ (pow z y) y)) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(a) * (t + -1.0);
	double t_2 = fma(x, (pow(a, t) / y), 0.0);
	double tmp;
	if (t_1 <= -1e+29) {
		tmp = t_2;
	} else if (t_1 <= -340.0) {
		tmp = (x * (((a - (a * b)) / a) / a)) / y;
	} else if (t_1 <= -42.0) {
		tmp = x / (y * exp(b));
	} else if (t_1 <= 1000.0) {
		tmp = x * (pow(z, y) / y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(a) * Float64(t + -1.0))
	t_2 = fma(x, Float64((a ^ t) / y), 0.0)
	tmp = 0.0
	if (t_1 <= -1e+29)
		tmp = t_2;
	elseif (t_1 <= -340.0)
		tmp = Float64(Float64(x * Float64(Float64(Float64(a - Float64(a * b)) / a) / a)) / y);
	elseif (t_1 <= -42.0)
		tmp = Float64(x / Float64(y * exp(b)));
	elseif (t_1 <= 1000.0)
		tmp = Float64(x * Float64((z ^ y) / y));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[Power[a, t], $MachinePrecision] / y), $MachinePrecision] + 0.0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+29], t$95$2, If[LessEqual[t$95$1, -340.0], N[(N[(x * N[(N[(N[(a - N[(a * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, -42.0], N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[(x * N[(N[Power[z, y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -9.99999999999999914e28 or 1e3 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y} + 0} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} + 0 \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}, 0\right)} \]

   5. Simplified 66.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, {a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{y}, 0\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{y}}, 0\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{y}}, 0\right) \]

      2. exp-to-pow N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{{a}^{\left(t - 1\right)}}}{y}, 0\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{{a}^{\left(t - 1\right)}}}{y}, 0\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}}{y}, 0\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\left(t + \color{blue}{-1}\right)}}{y}, 0\right) \]

      6. +-lowering-+.f64 84.1

         \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\color{blue}{\left(t + -1\right)}}}{y}, 0\right) \]

   8. Simplified 84.1%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{{a}^{\left(t + -1\right)}}{y}}, 0\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\color{blue}{t}}}{y}, 0\right) \]
   10. Step-by-step derivation

   11. Simplified 84.1%

       \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\color{blue}{t}}}{y}, 0\right) \]

   if -9.99999999999999914e28 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -340

   1. Initial program 94.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 90.3

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 90.3%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 59.6

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 59.6%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 61.6

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 61.6%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. div-sub N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{a} - \frac{b}{a}\right)}}{y} \]

       2. frac-sub N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 \cdot a - a \cdot b}{a \cdot a}}}{y} \]

       3. associate-/r* N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{\frac{1 \cdot a - a \cdot b}{a}}{a}}}{y} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{\frac{1 \cdot a - a \cdot b}{a}}{a}}}{y} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 \cdot a - a \cdot b}{a}}}{a}}{y} \]

       6. *-lft-identity N/A

          \[\leadsto \frac{x \cdot \frac{\frac{\color{blue}{a} - a \cdot b}{a}}{a}}{y} \]

       7. --lowering--.f64 N/A

          \[\leadsto \frac{x \cdot \frac{\frac{\color{blue}{a - a \cdot b}}{a}}{a}}{y} \]

       8. *-lowering-*.f64 68.9

          \[\leadsto \frac{x \cdot \frac{\frac{a - \color{blue}{a \cdot b}}{a}}{a}}{y} \]

   13. Applied egg-rr 68.9%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{\frac{a - a \cdot b}{a}}{a}}}{y} \]

   if -340 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -42

   1. Initial program 99.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 73.9

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 73.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 68.9

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 68.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{0 - b}}{y}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{e^{0 - b}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      5. exp-diff N/A

         \[\leadsto \frac{x}{\frac{y}{\color{blue}{\frac{e^{0}}{e^{b}}}}} \]

      6. 1-exp N/A

         \[\leadsto \frac{x}{\frac{y}{\frac{\color{blue}{1}}{e^{b}}}} \]

      7. associate-/r/ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{1} \cdot e^{b}}} \]

      8. /-rgt-identity N/A

         \[\leadsto \frac{x}{\color{blue}{y} \cdot e^{b}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      10. exp-lowering-exp.f64 68.9

          \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 68.9%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   if -42 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 1e3

   1. Initial program 99.3%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 84.7

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 84.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot {z}^{y}}}{y} \]

      3. pow-lowering-pow.f64 65.9

         \[\leadsto \frac{x \cdot \color{blue}{{z}^{y}}}{y} \]

   8. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y}} \cdot x \]

      5. pow-lowering-pow.f64 65.9

         \[\leadsto \frac{\color{blue}{{z}^{y}}}{y} \cdot x \]

   10. Applied egg-rr 65.9%

       \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t + -1\right) \leq -1 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{{a}^{t}}{y}, 0\right)\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq -340:\\ \;\;\;\;\frac{x \cdot \frac{\frac{a - a \cdot b}{a}}{a}}{y}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq -42:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 1000:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{{a}^{t}}{y}, 0\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log a \cdot \left(t + -1\right)\\ t_2 := \frac{x \cdot e^{\mathsf{fma}\left(\log a, t, 0\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+27}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -340:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y \cdot e^{b}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(y, \log z, 0\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (log a) (+ t -1.0)))
        (t_2 (/ (* x (exp (- (fma (log a) t 0.0) b))) y)))
   (if (<= t_1 -2e+27)
     t_2
     (if (<= t_1 -340.0)
       (/ (* x (pow a (+ t -1.0))) (* y (exp b)))
       (if (<= t_1 2e+137) (/ (* x (exp (- (fma y (log z) 0.0) b))) y) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(a) * (t + -1.0);
	double t_2 = (x * exp((fma(log(a), t, 0.0) - b))) / y;
	double tmp;
	if (t_1 <= -2e+27) {
		tmp = t_2;
	} else if (t_1 <= -340.0) {
		tmp = (x * pow(a, (t + -1.0))) / (y * exp(b));
	} else if (t_1 <= 2e+137) {
		tmp = (x * exp((fma(y, log(z), 0.0) - b))) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(a) * Float64(t + -1.0))
	t_2 = Float64(Float64(x * exp(Float64(fma(log(a), t, 0.0) - b))) / y)
	tmp = 0.0
	if (t_1 <= -2e+27)
		tmp = t_2;
	elseif (t_1 <= -340.0)
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / Float64(y * exp(b)));
	elseif (t_1 <= 2e+137)
		tmp = Float64(Float64(x * exp(Float64(fma(y, log(z), 0.0) - b))) / y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t + 0.0), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+27], t$95$2, If[LessEqual[t$95$1, -340.0], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+137], N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision] + 0.0), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -2e27 or 2.0000000000000001e137 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(t \cdot \log a + 0\right)} - b}}{y} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\left(\color{blue}{\log a \cdot t} + 0\right) - b}}{y} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

      4. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

      6. rem-exp-log 97.0

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t, 0\right) - b}}{y} \]

   5. Simplified 97.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

   if -2e27 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -340

   1. Initial program 94.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right) - b}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right) - b} \cdot x}}{y} \]

      2. exp-diff N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{e^{b}}} \cdot x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{e^{b}}}}{y} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)} \cdot x}{y \cdot e^{b}}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{\log a \cdot \left(t - 1\right)} \cdot x}}{y \cdot e^{b}} \]

      7. exp-prod N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(e^{\log a}\right)}^{\left(t - 1\right)}} \cdot x}{y \cdot e^{b}} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{{\color{blue}{a}}^{\left(t - 1\right)} \cdot x}{y \cdot e^{b}} \]

      10. sub-neg N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}} \cdot x}{y \cdot e^{b}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{{a}^{\left(t + \color{blue}{-1}\right)} \cdot x}{y \cdot e^{b}} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{{a}^{\color{blue}{\left(t + -1\right)}} \cdot x}{y \cdot e^{b}} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{\color{blue}{y \cdot e^{b}}} \]

      14. exp-lowering-exp.f64 81.6

          \[\leadsto \frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot \color{blue}{e^{b}}} \]

   5. Simplified 81.6%

      \[\leadsto \color{blue}{\frac{{a}^{\left(t + -1\right)} \cdot x}{y \cdot e^{b}}} \]

   if -340 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 2.0000000000000001e137

   1. Initial program 99.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 86.2

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 86.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t + -1\right) \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(\log a, t, 0\right) - b}}{y}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq -340:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y \cdot e^{b}}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(y, \log z, 0\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(\log a, t, 0\right) - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log a \cdot \left(t + -1\right)\\ t_2 := \frac{x \cdot e^{\mathsf{fma}\left(\log a, t, 0\right) - b}}{y}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+29}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(y, \log z, 0\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (log a) (+ t -1.0)))
        (t_2 (/ (* x (exp (- (fma (log a) t 0.0) b))) y)))
   (if (<= t_1 -1e+29)
     t_2
     (if (<= t_1 2e+137) (/ (* x (exp (- (fma y (log z) 0.0) b))) y) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(a) * (t + -1.0);
	double t_2 = (x * exp((fma(log(a), t, 0.0) - b))) / y;
	double tmp;
	if (t_1 <= -1e+29) {
		tmp = t_2;
	} else if (t_1 <= 2e+137) {
		tmp = (x * exp((fma(y, log(z), 0.0) - b))) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(a) * Float64(t + -1.0))
	t_2 = Float64(Float64(x * exp(Float64(fma(log(a), t, 0.0) - b))) / y)
	tmp = 0.0
	if (t_1 <= -1e+29)
		tmp = t_2;
	elseif (t_1 <= 2e+137)
		tmp = Float64(Float64(x * exp(Float64(fma(y, log(z), 0.0) - b))) / y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t + 0.0), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+29], t$95$2, If[LessEqual[t$95$1, 2e+137], N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision] + 0.0), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -9.99999999999999914e28 or 2.0000000000000001e137 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(t \cdot \log a + 0\right)} - b}}{y} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\left(\color{blue}{\log a \cdot t} + 0\right) - b}}{y} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

      4. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

      6. rem-exp-log 96.9

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t, 0\right) - b}}{y} \]

   5. Simplified 96.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

   if -9.99999999999999914e28 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 2.0000000000000001e137

   1. Initial program 98.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 81.7

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 81.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t + -1\right) \leq -1 \cdot 10^{+29}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(\log a, t, 0\right) - b}}{y}\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 2 \cdot 10^{+137}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(y, \log z, 0\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(\log a, t, 0\right) - b}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log a \cdot \left(t + -1\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{{a}^{t}}{y}, 0\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+191}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(y, \log z, 0\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (log a) (+ t -1.0))))
   (if (<= t_1 -5e+94)
     (fma x (/ (pow a t) y) 0.0)
     (if (<= t_1 1e+191)
       (/ (* x (exp (- (fma y (log z) 0.0) b))) y)
       (/ (* x (pow a (+ t -1.0))) y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(a) * (t + -1.0);
	double tmp;
	if (t_1 <= -5e+94) {
		tmp = fma(x, (pow(a, t) / y), 0.0);
	} else if (t_1 <= 1e+191) {
		tmp = (x * exp((fma(y, log(z), 0.0) - b))) / y;
	} else {
		tmp = (x * pow(a, (t + -1.0))) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(a) * Float64(t + -1.0))
	tmp = 0.0
	if (t_1 <= -5e+94)
		tmp = fma(x, Float64((a ^ t) / y), 0.0);
	elseif (t_1 <= 1e+191)
		tmp = Float64(Float64(x * exp(Float64(fma(y, log(z), 0.0) - b))) / y);
	else
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+94], N[(x * N[(N[Power[a, t], $MachinePrecision] / y), $MachinePrecision] + 0.0), $MachinePrecision], If[LessEqual[t$95$1, 1e+191], N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision] + 0.0), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < -5.0000000000000001e94

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y} + 0} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} + 0 \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}, 0\right)} \]

   5. Simplified 71.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, {a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{y}, 0\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{y}}, 0\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{y}}, 0\right) \]

      2. exp-to-pow N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{{a}^{\left(t - 1\right)}}}{y}, 0\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{{a}^{\left(t - 1\right)}}}{y}, 0\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}}{y}, 0\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\left(t + \color{blue}{-1}\right)}}{y}, 0\right) \]

      6. +-lowering-+.f64 90.6

         \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\color{blue}{\left(t + -1\right)}}}{y}, 0\right) \]

   8. Simplified 90.6%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{{a}^{\left(t + -1\right)}}{y}}, 0\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\color{blue}{t}}}{y}, 0\right) \]
   10. Step-by-step derivation

   11. Simplified 90.6%

       \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\color{blue}{t}}}{y}, 0\right) \]

   if -5.0000000000000001e94 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a)) < 1.00000000000000007e191

   1. Initial program 98.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 80.6

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 80.6%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   if 1.00000000000000007e191 < (*.f64 (-.f64 t #s(literal 1 binary64)) (log.f64 a))

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 100.0

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 100.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\log a \cdot \left(t - 1\right)}}}{y} \]

      3. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \color{blue}{{a}^{\left(t - 1\right)}}}{y} \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{{a}^{\left(t - 1\right)}}}{y} \]

      5. sub-neg N/A

         \[\leadsto \frac{x \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}}{y} \]

      6. metadata-eval N/A

         \[\leadsto \frac{x \cdot {a}^{\left(t + \color{blue}{-1}\right)}}{y} \]

      7. +-lowering-+.f64 97.0

         \[\leadsto \frac{x \cdot {a}^{\color{blue}{\left(t + -1\right)}}}{y} \]

   8. Simplified 97.0%

      \[\leadsto \color{blue}{\frac{x \cdot {a}^{\left(t + -1\right)}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \cdot \left(t + -1\right) \leq -5 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{{a}^{t}}{y}, 0\right)\\ \mathbf{elif}\;\log a \cdot \left(t + -1\right) \leq 10^{+191}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(y, \log z, 0\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot e^{\mathsf{fma}\left(y, \log z, 0\right) - b}}{y}\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(\log a, t + -1, 0\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* x (exp (- (fma y (log z) 0.0) b))) y)))
   (if (<= y -3.7e+84)
     t_1
     (if (<= y 7.5e-10)
       (/ (* x (exp (- (fma (log a) (+ t -1.0) 0.0) b))) y)
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * exp((fma(y, log(z), 0.0) - b))) / y;
	double tmp;
	if (y <= -3.7e+84) {
		tmp = t_1;
	} else if (y <= 7.5e-10) {
		tmp = (x * exp((fma(log(a), (t + -1.0), 0.0) - b))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * exp(Float64(fma(y, log(z), 0.0) - b))) / y)
	tmp = 0.0
	if (y <= -3.7e+84)
		tmp = t_1;
	elseif (y <= 7.5e-10)
		tmp = Float64(Float64(x * exp(Float64(fma(log(a), Float64(t + -1.0), 0.0) - b))) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision] + 0.0), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -3.7e+84], t$95$1, If[LessEqual[y, 7.5e-10], N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * N[(t + -1.0), $MachinePrecision] + 0.0), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7e84 or 7.49999999999999995e-10 < y

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 84.9

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 84.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   if -3.7e84 < y < 7.49999999999999995e-10

   1. Initial program 98.4%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 96.5

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 96.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 87.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -600:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(y, \log z, 0\right) - b}}{y}\\ \mathbf{elif}\;b \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(x, {a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{y}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\mathsf{fma}\left(\log a, t, 0\right) - b}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -600.0)
   (/ (* x (exp (- (fma y (log z) 0.0) b))) y)
   (if (<= b 1000.0)
     (fma x (* (pow a (+ t -1.0)) (/ (pow z y) y)) 0.0)
     (/ (* x (exp (- (fma (log a) t 0.0) b))) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -600.0) {
		tmp = (x * exp((fma(y, log(z), 0.0) - b))) / y;
	} else if (b <= 1000.0) {
		tmp = fma(x, (pow(a, (t + -1.0)) * (pow(z, y) / y)), 0.0);
	} else {
		tmp = (x * exp((fma(log(a), t, 0.0) - b))) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -600.0)
		tmp = Float64(Float64(x * exp(Float64(fma(y, log(z), 0.0) - b))) / y);
	elseif (b <= 1000.0)
		tmp = fma(x, Float64((a ^ Float64(t + -1.0)) * Float64((z ^ y) / y)), 0.0);
	else
		tmp = Float64(Float64(x * exp(Float64(fma(log(a), t, 0.0) - b))) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -600.0], N[(N[(x * N[Exp[N[(N[(y * N[Log[z], $MachinePrecision] + 0.0), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 1000.0], N[(x * N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[z, y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision], N[(N[(x * N[Exp[N[(N[(N[Log[a], $MachinePrecision] * t + 0.0), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -600

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 98.2

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 98.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   if -600 < b < 1e3

   1. Initial program 98.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y} + 0} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} + 0 \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}, 0\right)} \]

   5. Simplified 83.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, {a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{y}, 0\right)} \]

   if 1e3 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{t \cdot \log a} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(t \cdot \log a + 0\right)} - b}}{y} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot e^{\left(\color{blue}{\log a \cdot t} + 0\right) - b}}{y} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]

      4. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t, 0\right) - b}}{y} \]

      6. rem-exp-log 91.6

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t, 0\right) - b}}{y} \]

   5. Simplified 91.6%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t, 0\right)} - b}}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y \cdot e^{b}}\\ \mathbf{if}\;b \leq -1000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 6:\\ \;\;\;\;\frac{\frac{{a}^{t} \cdot \left(x \cdot \left(1 - b\right)\right)}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* y (exp b)))))
   (if (<= b -1000.0)
     t_1
     (if (<= b 6.0) (/ (/ (* (pow a t) (* x (- 1.0 b))) a) y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * exp(b));
	double tmp;
	if (b <= -1000.0) {
		tmp = t_1;
	} else if (b <= 6.0) {
		tmp = ((pow(a, t) * (x * (1.0 - b))) / a) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (y * exp(b))
    if (b <= (-1000.0d0)) then
        tmp = t_1
    else if (b <= 6.0d0) then
        tmp = (((a ** t) * (x * (1.0d0 - b))) / a) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * Math.exp(b));
	double tmp;
	if (b <= -1000.0) {
		tmp = t_1;
	} else if (b <= 6.0) {
		tmp = ((Math.pow(a, t) * (x * (1.0 - b))) / a) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (y * math.exp(b))
	tmp = 0
	if b <= -1000.0:
		tmp = t_1
	elif b <= 6.0:
		tmp = ((math.pow(a, t) * (x * (1.0 - b))) / a) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(y * exp(b)))
	tmp = 0.0
	if (b <= -1000.0)
		tmp = t_1;
	elseif (b <= 6.0)
		tmp = Float64(Float64(Float64((a ^ t) * Float64(x * Float64(1.0 - b))) / a) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (y * exp(b));
	tmp = 0.0;
	if (b <= -1000.0)
		tmp = t_1;
	elseif (b <= 6.0)
		tmp = (((a ^ t) * (x * (1.0 - b))) / a) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1000.0], t$95$1, If[LessEqual[b, 6.0], N[(N[(N[(N[Power[a, t], $MachinePrecision] * N[(x * N[(1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1e3 or 6 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 91.3

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 91.3%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 83.5

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 83.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{0 - b}}{y}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{e^{0 - b}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      5. exp-diff N/A

         \[\leadsto \frac{x}{\frac{y}{\color{blue}{\frac{e^{0}}{e^{b}}}}} \]

      6. 1-exp N/A

         \[\leadsto \frac{x}{\frac{y}{\frac{\color{blue}{1}}{e^{b}}}} \]

      7. associate-/r/ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{1} \cdot e^{b}}} \]

      8. /-rgt-identity N/A

         \[\leadsto \frac{x}{\color{blue}{y} \cdot e^{b}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      10. exp-lowering-exp.f64 83.5

          \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 83.5%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   if -1e3 < b < 6

   1. Initial program 98.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 73.2

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 73.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]
   6. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{e^{\log a \cdot \left(t + -1\right) + 0}}{e^{b}}}}{y} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot \frac{e^{\color{blue}{\log a \cdot \left(t + -1\right)}}}{e^{b}}}{y} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{a}^{\left(t + -1\right)}}}{e^{b}}}{y} \]

      4. unpow-prod-up N/A

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{a}^{t} \cdot {a}^{-1}}}{e^{b}}}{y} \]

      5. inv-pow N/A

         \[\leadsto \frac{x \cdot \frac{{a}^{t} \cdot \color{blue}{\frac{1}{a}}}{e^{b}}}{y} \]

      6. associate-/l* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left({a}^{t} \cdot \frac{\frac{1}{a}}{e^{b}}\right)}}{y} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left({a}^{t} \cdot \frac{\frac{1}{a}}{e^{b}}\right)}}{y} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{a}^{t}} \cdot \frac{\frac{1}{a}}{e^{b}}\right)}{y} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \color{blue}{\frac{\frac{1}{a}}{e^{b}}}\right)}{y} \]

      10. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \frac{\frac{\color{blue}{-1 \cdot -1}}{a}}{e^{b}}\right)}{y} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \frac{\color{blue}{\frac{-1 \cdot -1}{a}}}{e^{b}}\right)}{y} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \frac{\frac{\color{blue}{1}}{a}}{e^{b}}\right)}{y} \]

      13. exp-lowering-exp.f64 74.1

          \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \frac{\frac{1}{a}}{\color{blue}{e^{b}}}\right)}{y} \]

   7. Applied egg-rr 74.1%

      \[\leadsto \frac{x \cdot \color{blue}{\left({a}^{t} \cdot \frac{\frac{1}{a}}{e^{b}}\right)}}{y} \]

   8. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(-1 \cdot \frac{b \cdot {a}^{t}}{a} + \frac{{a}^{t}}{a}\right)}}{y} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{{a}^{t}}{a} + -1 \cdot \frac{b \cdot {a}^{t}}{a}\right)}}{y} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x \cdot \left(\frac{{a}^{t}}{a} + \color{blue}{\left(\mathsf{neg}\left(\frac{b \cdot {a}^{t}}{a}\right)\right)}\right)}{y} \]

      3. unsub-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{{a}^{t}}{a} - \frac{b \cdot {a}^{t}}{a}\right)}}{y} \]

   10. Simplified 74.0%

       \[\leadsto \frac{x \cdot \color{blue}{\left({a}^{\left(-1 + t\right)} \cdot \left(1 - b\right)\right)}}{y} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot {a}^{\left(-1 + t\right)}\right) \cdot \left(1 - b\right)}}{y} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left({a}^{\left(-1 + t\right)} \cdot x\right)} \cdot \left(1 - b\right)}{y} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\color{blue}{{a}^{\left(-1 + t\right)} \cdot \left(x \cdot \left(1 - b\right)\right)}}{y} \]

       4. unpow-prod-up N/A

          \[\leadsto \frac{\color{blue}{\left({a}^{-1} \cdot {a}^{t}\right)} \cdot \left(x \cdot \left(1 - b\right)\right)}{y} \]

       5. inv-pow N/A

          \[\leadsto \frac{\left(\color{blue}{\frac{1}{a}} \cdot {a}^{t}\right) \cdot \left(x \cdot \left(1 - b\right)\right)}{y} \]

       6. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left({a}^{t} \cdot \frac{1}{a}\right)} \cdot \left(x \cdot \left(1 - b\right)\right)}{y} \]

       7. un-div-inv N/A

          \[\leadsto \frac{\color{blue}{\frac{{a}^{t}}{a}} \cdot \left(x \cdot \left(1 - b\right)\right)}{y} \]

       8. associate-*l/ N/A

          \[\leadsto \frac{\color{blue}{\frac{{a}^{t} \cdot \left(x \cdot \left(1 - b\right)\right)}{a}}}{y} \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{{a}^{t} \cdot \left(x \cdot \left(1 - b\right)\right)}{a}}}{y} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \frac{\frac{\color{blue}{{a}^{t} \cdot \left(x \cdot \left(1 - b\right)\right)}}{a}}{y} \]

       11. pow-lowering-pow.f64 N/A

           \[\leadsto \frac{\frac{\color{blue}{{a}^{t}} \cdot \left(x \cdot \left(1 - b\right)\right)}{a}}{y} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \frac{\frac{{a}^{t} \cdot \color{blue}{\left(x \cdot \left(1 - b\right)\right)}}{a}}{y} \]

       13. --lowering--.f64 74.2

           \[\leadsto \frac{\frac{{a}^{t} \cdot \left(x \cdot \color{blue}{\left(1 - b\right)}\right)}{a}}{y} \]

   12. Applied egg-rr 74.2%

       \[\leadsto \frac{\color{blue}{\frac{{a}^{t} \cdot \left(x \cdot \left(1 - b\right)\right)}{a}}}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y \cdot e^{b}}\\ \mathbf{if}\;b \leq -105000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 6.8:\\ \;\;\;\;\frac{x \cdot \left({a}^{t} \cdot \frac{1 - b}{a}\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* y (exp b)))))
   (if (<= b -105000.0)
     t_1
     (if (<= b 6.8) (/ (* x (* (pow a t) (/ (- 1.0 b) a))) y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * exp(b));
	double tmp;
	if (b <= -105000.0) {
		tmp = t_1;
	} else if (b <= 6.8) {
		tmp = (x * (pow(a, t) * ((1.0 - b) / a))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (y * exp(b))
    if (b <= (-105000.0d0)) then
        tmp = t_1
    else if (b <= 6.8d0) then
        tmp = (x * ((a ** t) * ((1.0d0 - b) / a))) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * Math.exp(b));
	double tmp;
	if (b <= -105000.0) {
		tmp = t_1;
	} else if (b <= 6.8) {
		tmp = (x * (Math.pow(a, t) * ((1.0 - b) / a))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (y * math.exp(b))
	tmp = 0
	if b <= -105000.0:
		tmp = t_1
	elif b <= 6.8:
		tmp = (x * (math.pow(a, t) * ((1.0 - b) / a))) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(y * exp(b)))
	tmp = 0.0
	if (b <= -105000.0)
		tmp = t_1;
	elseif (b <= 6.8)
		tmp = Float64(Float64(x * Float64((a ^ t) * Float64(Float64(1.0 - b) / a))) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (y * exp(b));
	tmp = 0.0;
	if (b <= -105000.0)
		tmp = t_1;
	elseif (b <= 6.8)
		tmp = (x * ((a ^ t) * ((1.0 - b) / a))) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -105000.0], t$95$1, If[LessEqual[b, 6.8], N[(N[(x * N[(N[Power[a, t], $MachinePrecision] * N[(N[(1.0 - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -105000 or 6.79999999999999982 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 91.3

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 91.3%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 83.5

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 83.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{0 - b}}{y}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{e^{0 - b}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      5. exp-diff N/A

         \[\leadsto \frac{x}{\frac{y}{\color{blue}{\frac{e^{0}}{e^{b}}}}} \]

      6. 1-exp N/A

         \[\leadsto \frac{x}{\frac{y}{\frac{\color{blue}{1}}{e^{b}}}} \]

      7. associate-/r/ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{1} \cdot e^{b}}} \]

      8. /-rgt-identity N/A

         \[\leadsto \frac{x}{\color{blue}{y} \cdot e^{b}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      10. exp-lowering-exp.f64 83.5

          \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 83.5%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   if -105000 < b < 6.79999999999999982

   1. Initial program 98.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 73.2

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 73.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]
   6. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{e^{\log a \cdot \left(t + -1\right) + 0}}{e^{b}}}}{y} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot \frac{e^{\color{blue}{\log a \cdot \left(t + -1\right)}}}{e^{b}}}{y} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{a}^{\left(t + -1\right)}}}{e^{b}}}{y} \]

      4. unpow-prod-up N/A

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{a}^{t} \cdot {a}^{-1}}}{e^{b}}}{y} \]

      5. inv-pow N/A

         \[\leadsto \frac{x \cdot \frac{{a}^{t} \cdot \color{blue}{\frac{1}{a}}}{e^{b}}}{y} \]

      6. associate-/l* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left({a}^{t} \cdot \frac{\frac{1}{a}}{e^{b}}\right)}}{y} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left({a}^{t} \cdot \frac{\frac{1}{a}}{e^{b}}\right)}}{y} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{a}^{t}} \cdot \frac{\frac{1}{a}}{e^{b}}\right)}{y} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \color{blue}{\frac{\frac{1}{a}}{e^{b}}}\right)}{y} \]

      10. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \frac{\frac{\color{blue}{-1 \cdot -1}}{a}}{e^{b}}\right)}{y} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \frac{\color{blue}{\frac{-1 \cdot -1}{a}}}{e^{b}}\right)}{y} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \frac{\frac{\color{blue}{1}}{a}}{e^{b}}\right)}{y} \]

      13. exp-lowering-exp.f64 74.1

          \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \frac{\frac{1}{a}}{\color{blue}{e^{b}}}\right)}{y} \]

   7. Applied egg-rr 74.1%

      \[\leadsto \frac{x \cdot \color{blue}{\left({a}^{t} \cdot \frac{\frac{1}{a}}{e^{b}}\right)}}{y} \]

   8. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \color{blue}{\left(-1 \cdot \frac{b}{a} + \frac{1}{a}\right)}\right)}{y} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{b}{a}\right)\right)} + \frac{1}{a}\right)\right)}{y} \]

      2. +-commutative N/A

         \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \color{blue}{\left(\frac{1}{a} + \left(\mathsf{neg}\left(\frac{b}{a}\right)\right)\right)}\right)}{y} \]

      3. sub-neg N/A

         \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \color{blue}{\left(\frac{1}{a} - \frac{b}{a}\right)}\right)}{y} \]

      4. div-sub N/A

         \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \color{blue}{\frac{1 - b}{a}}\right)}{y} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \color{blue}{\frac{1 - b}{a}}\right)}{y} \]

      6. --lowering--.f64 74.1

         \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \frac{\color{blue}{1 - b}}{a}\right)}{y} \]

   10. Simplified 74.1%

       \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \color{blue}{\frac{1 - b}{a}}\right)}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y \cdot e^{b}}\\ \mathbf{if}\;b \leq -800:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 1.55:\\ \;\;\;\;\frac{x \cdot \left({a}^{\left(t + -1\right)} \cdot \left(1 - b\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* y (exp b)))))
   (if (<= b -800.0)
     t_1
     (if (<= b 1.55) (/ (* x (* (pow a (+ t -1.0)) (- 1.0 b))) y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * exp(b));
	double tmp;
	if (b <= -800.0) {
		tmp = t_1;
	} else if (b <= 1.55) {
		tmp = (x * (pow(a, (t + -1.0)) * (1.0 - b))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (y * exp(b))
    if (b <= (-800.0d0)) then
        tmp = t_1
    else if (b <= 1.55d0) then
        tmp = (x * ((a ** (t + (-1.0d0))) * (1.0d0 - b))) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * Math.exp(b));
	double tmp;
	if (b <= -800.0) {
		tmp = t_1;
	} else if (b <= 1.55) {
		tmp = (x * (Math.pow(a, (t + -1.0)) * (1.0 - b))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (y * math.exp(b))
	tmp = 0
	if b <= -800.0:
		tmp = t_1
	elif b <= 1.55:
		tmp = (x * (math.pow(a, (t + -1.0)) * (1.0 - b))) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(y * exp(b)))
	tmp = 0.0
	if (b <= -800.0)
		tmp = t_1;
	elseif (b <= 1.55)
		tmp = Float64(Float64(x * Float64((a ^ Float64(t + -1.0)) * Float64(1.0 - b))) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (y * exp(b));
	tmp = 0.0;
	if (b <= -800.0)
		tmp = t_1;
	elseif (b <= 1.55)
		tmp = (x * ((a ^ (t + -1.0)) * (1.0 - b))) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -800.0], t$95$1, If[LessEqual[b, 1.55], N[(N[(x * N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] * N[(1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -800 or 1.55000000000000004 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 91.3

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 91.3%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 83.5

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 83.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{0 - b}}{y}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{e^{0 - b}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      5. exp-diff N/A

         \[\leadsto \frac{x}{\frac{y}{\color{blue}{\frac{e^{0}}{e^{b}}}}} \]

      6. 1-exp N/A

         \[\leadsto \frac{x}{\frac{y}{\frac{\color{blue}{1}}{e^{b}}}} \]

      7. associate-/r/ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{1} \cdot e^{b}}} \]

      8. /-rgt-identity N/A

         \[\leadsto \frac{x}{\color{blue}{y} \cdot e^{b}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      10. exp-lowering-exp.f64 83.5

          \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 83.5%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   if -800 < b < 1.55000000000000004

   1. Initial program 98.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 73.2

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 73.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]
   6. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{x \cdot \color{blue}{\frac{e^{\log a \cdot \left(t + -1\right) + 0}}{e^{b}}}}{y} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot \frac{e^{\color{blue}{\log a \cdot \left(t + -1\right)}}}{e^{b}}}{y} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{a}^{\left(t + -1\right)}}}{e^{b}}}{y} \]

      4. unpow-prod-up N/A

         \[\leadsto \frac{x \cdot \frac{\color{blue}{{a}^{t} \cdot {a}^{-1}}}{e^{b}}}{y} \]

      5. inv-pow N/A

         \[\leadsto \frac{x \cdot \frac{{a}^{t} \cdot \color{blue}{\frac{1}{a}}}{e^{b}}}{y} \]

      6. associate-/l* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left({a}^{t} \cdot \frac{\frac{1}{a}}{e^{b}}\right)}}{y} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left({a}^{t} \cdot \frac{\frac{1}{a}}{e^{b}}\right)}}{y} \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{{a}^{t}} \cdot \frac{\frac{1}{a}}{e^{b}}\right)}{y} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \color{blue}{\frac{\frac{1}{a}}{e^{b}}}\right)}{y} \]

      10. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \frac{\frac{\color{blue}{-1 \cdot -1}}{a}}{e^{b}}\right)}{y} \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \frac{\color{blue}{\frac{-1 \cdot -1}{a}}}{e^{b}}\right)}{y} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \frac{\frac{\color{blue}{1}}{a}}{e^{b}}\right)}{y} \]

      13. exp-lowering-exp.f64 74.1

          \[\leadsto \frac{x \cdot \left({a}^{t} \cdot \frac{\frac{1}{a}}{\color{blue}{e^{b}}}\right)}{y} \]

   7. Applied egg-rr 74.1%

      \[\leadsto \frac{x \cdot \color{blue}{\left({a}^{t} \cdot \frac{\frac{1}{a}}{e^{b}}\right)}}{y} \]

   8. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(-1 \cdot \frac{b \cdot {a}^{t}}{a} + \frac{{a}^{t}}{a}\right)}}{y} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{{a}^{t}}{a} + -1 \cdot \frac{b \cdot {a}^{t}}{a}\right)}}{y} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x \cdot \left(\frac{{a}^{t}}{a} + \color{blue}{\left(\mathsf{neg}\left(\frac{b \cdot {a}^{t}}{a}\right)\right)}\right)}{y} \]

      3. unsub-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{{a}^{t}}{a} - \frac{b \cdot {a}^{t}}{a}\right)}}{y} \]

   10. Simplified 74.0%

       \[\leadsto \frac{x \cdot \color{blue}{\left({a}^{\left(-1 + t\right)} \cdot \left(1 - b\right)\right)}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -800:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;b \leq 1.55:\\ \;\;\;\;\frac{x \cdot \left({a}^{\left(t + -1\right)} \cdot \left(1 - b\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1 - b}{a}\\ t_2 := \frac{x}{y \cdot e^{b}}\\ t_3 := \frac{1}{a} \cdot \left(1 + b\right)\\ \mathbf{if}\;b \leq -0.051:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-296}:\\ \;\;\;\;\frac{t\_1}{\frac{y}{x}}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{x \cdot \left(t\_1 \cdot t\_3\right)}{t\_3}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- 1.0 b) a))
        (t_2 (/ x (* y (exp b))))
        (t_3 (* (/ 1.0 a) (+ 1.0 b))))
   (if (<= b -0.051)
     t_2
     (if (<= b 3.2e-296)
       (/ t_1 (/ y x))
       (if (<= b 6.5e-31) (/ (/ (* x (* t_1 t_3)) t_3) y) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 - b) / a;
	double t_2 = x / (y * exp(b));
	double t_3 = (1.0 / a) * (1.0 + b);
	double tmp;
	if (b <= -0.051) {
		tmp = t_2;
	} else if (b <= 3.2e-296) {
		tmp = t_1 / (y / x);
	} else if (b <= 6.5e-31) {
		tmp = ((x * (t_1 * t_3)) / t_3) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (1.0d0 - b) / a
    t_2 = x / (y * exp(b))
    t_3 = (1.0d0 / a) * (1.0d0 + b)
    if (b <= (-0.051d0)) then
        tmp = t_2
    else if (b <= 3.2d-296) then
        tmp = t_1 / (y / x)
    else if (b <= 6.5d-31) then
        tmp = ((x * (t_1 * t_3)) / t_3) / y
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 - b) / a;
	double t_2 = x / (y * Math.exp(b));
	double t_3 = (1.0 / a) * (1.0 + b);
	double tmp;
	if (b <= -0.051) {
		tmp = t_2;
	} else if (b <= 3.2e-296) {
		tmp = t_1 / (y / x);
	} else if (b <= 6.5e-31) {
		tmp = ((x * (t_1 * t_3)) / t_3) / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (1.0 - b) / a
	t_2 = x / (y * math.exp(b))
	t_3 = (1.0 / a) * (1.0 + b)
	tmp = 0
	if b <= -0.051:
		tmp = t_2
	elif b <= 3.2e-296:
		tmp = t_1 / (y / x)
	elif b <= 6.5e-31:
		tmp = ((x * (t_1 * t_3)) / t_3) / y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.0 - b) / a)
	t_2 = Float64(x / Float64(y * exp(b)))
	t_3 = Float64(Float64(1.0 / a) * Float64(1.0 + b))
	tmp = 0.0
	if (b <= -0.051)
		tmp = t_2;
	elseif (b <= 3.2e-296)
		tmp = Float64(t_1 / Float64(y / x));
	elseif (b <= 6.5e-31)
		tmp = Float64(Float64(Float64(x * Float64(t_1 * t_3)) / t_3) / y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (1.0 - b) / a;
	t_2 = x / (y * exp(b));
	t_3 = (1.0 / a) * (1.0 + b);
	tmp = 0.0;
	if (b <= -0.051)
		tmp = t_2;
	elseif (b <= 3.2e-296)
		tmp = t_1 / (y / x);
	elseif (b <= 6.5e-31)
		tmp = ((x * (t_1 * t_3)) / t_3) / y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - b), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 / a), $MachinePrecision] * N[(1.0 + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.051], t$95$2, If[LessEqual[b, 3.2e-296], N[(t$95$1 / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e-31], N[(N[(N[(x * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / y), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.0509999999999999967 or 6.49999999999999967e-31 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 90.3

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 90.3%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 80.5

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 80.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{0 - b}}{y}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{e^{0 - b}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      5. exp-diff N/A

         \[\leadsto \frac{x}{\frac{y}{\color{blue}{\frac{e^{0}}{e^{b}}}}} \]

      6. 1-exp N/A

         \[\leadsto \frac{x}{\frac{y}{\frac{\color{blue}{1}}{e^{b}}}} \]

      7. associate-/r/ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{1} \cdot e^{b}}} \]

      8. /-rgt-identity N/A

         \[\leadsto \frac{x}{\color{blue}{y} \cdot e^{b}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      10. exp-lowering-exp.f64 80.5

          \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 80.5%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   if -0.0509999999999999967 < b < 3.20000000000000013e-296

   1. Initial program 97.7%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 74.7

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 74.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 75.2

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 75.2%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 40.7

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 40.7%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{1 - b}{a} \cdot x}}{y} \]

       2. associate-/l* N/A

          \[\leadsto \color{blue}{\frac{1 - b}{a} \cdot \frac{x}{y}} \]

       3. clear-num N/A

          \[\leadsto \frac{1 - b}{a} \cdot \color{blue}{\frac{1}{\frac{y}{x}}} \]

       4. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{\frac{1 - b}{a}}{\frac{y}{x}}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1 - b}{a}}{\frac{y}{x}}} \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1 - b}{a}}}{\frac{y}{x}} \]

       7. --lowering--.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{1 - b}}{a}}{\frac{y}{x}} \]

       8. /-lowering-/.f64 45.7

          \[\leadsto \frac{\frac{1 - b}{a}}{\color{blue}{\frac{y}{x}}} \]

   13. Applied egg-rr 45.7%

       \[\leadsto \color{blue}{\frac{\frac{1 - b}{a}}{\frac{y}{x}}} \]

   if 3.20000000000000013e-296 < b < 6.49999999999999967e-31

   1. Initial program 98.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 71.6

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 71.6%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 72.8

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 72.8%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 43.7

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 43.7%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{1 - b}{a} \cdot x}}{y} \]

       2. div-sub N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{b}{a}\right)} \cdot x}{y} \]

       3. flip-- N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{a} \cdot \frac{1}{a} - \frac{b}{a} \cdot \frac{b}{a}}{\frac{1}{a} + \frac{b}{a}}} \cdot x}{y} \]

       4. associate-*l/ N/A

          \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{a} \cdot \frac{1}{a} - \frac{b}{a} \cdot \frac{b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}}{y} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{a} \cdot \frac{1}{a} - \frac{b}{a} \cdot \frac{b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}}{y} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{a} \cdot \frac{1}{a} - \frac{b}{a} \cdot \frac{b}{a}\right) \cdot x}}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       7. difference-of-squares N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(\left(\frac{1}{a} + \frac{b}{a}\right) \cdot \left(\frac{1}{a} - \frac{b}{a}\right)\right)} \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       8. div-sub N/A

          \[\leadsto \frac{\frac{\left(\left(\frac{1}{a} + \frac{b}{a}\right) \cdot \color{blue}{\frac{1 - b}{a}}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(\left(\frac{1}{a} + \frac{b}{a}\right) \cdot \frac{1 - b}{a}\right)} \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       10. div-inv N/A

           \[\leadsto \frac{\frac{\left(\left(\color{blue}{1 \cdot \frac{1}{a}} + \frac{b}{a}\right) \cdot \frac{1 - b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       11. div-inv N/A

           \[\leadsto \frac{\frac{\left(\left(1 \cdot \frac{1}{a} + \color{blue}{b \cdot \frac{1}{a}}\right) \cdot \frac{1 - b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       12. distribute-rgt-out N/A

           \[\leadsto \frac{\frac{\left(\color{blue}{\left(\frac{1}{a} \cdot \left(1 + b\right)\right)} \cdot \frac{1 - b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{\frac{\left(\color{blue}{\left(\frac{1}{a} \cdot \left(1 + b\right)\right)} \cdot \frac{1 - b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \frac{\frac{\left(\left(\color{blue}{\frac{1}{a}} \cdot \left(1 + b\right)\right) \cdot \frac{1 - b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       15. +-lowering-+.f64 N/A

           \[\leadsto \frac{\frac{\left(\left(\frac{1}{a} \cdot \color{blue}{\left(1 + b\right)}\right) \cdot \frac{1 - b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \frac{\frac{\left(\left(\frac{1}{a} \cdot \left(1 + b\right)\right) \cdot \color{blue}{\frac{1 - b}{a}}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       17. --lowering--.f64 N/A

           \[\leadsto \frac{\frac{\left(\left(\frac{1}{a} \cdot \left(1 + b\right)\right) \cdot \frac{\color{blue}{1 - b}}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

   13. Applied egg-rr 48.5%

       \[\leadsto \frac{\color{blue}{\frac{\left(\left(\frac{1}{a} \cdot \left(1 + b\right)\right) \cdot \frac{1 - b}{a}\right) \cdot x}{\frac{1}{a} \cdot \left(1 + b\right)}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.051:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-296}:\\ \;\;\;\;\frac{\frac{1 - b}{a}}{\frac{y}{x}}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{x \cdot \left(\frac{1 - b}{a} \cdot \left(\frac{1}{a} \cdot \left(1 + b\right)\right)\right)}{\frac{1}{a} \cdot \left(1 + b\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y \cdot e^{b}}\\ \mathbf{if}\;b \leq -900:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.75 \cdot 10^{+14}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* y (exp b)))))
   (if (<= b -900.0)
     t_1
     (if (<= b 2.75e+14) (/ (* x (pow a (+ t -1.0))) y) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * exp(b));
	double tmp;
	if (b <= -900.0) {
		tmp = t_1;
	} else if (b <= 2.75e+14) {
		tmp = (x * pow(a, (t + -1.0))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (y * exp(b))
    if (b <= (-900.0d0)) then
        tmp = t_1
    else if (b <= 2.75d+14) then
        tmp = (x * (a ** (t + (-1.0d0)))) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * Math.exp(b));
	double tmp;
	if (b <= -900.0) {
		tmp = t_1;
	} else if (b <= 2.75e+14) {
		tmp = (x * Math.pow(a, (t + -1.0))) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (y * math.exp(b))
	tmp = 0
	if b <= -900.0:
		tmp = t_1
	elif b <= 2.75e+14:
		tmp = (x * math.pow(a, (t + -1.0))) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(y * exp(b)))
	tmp = 0.0
	if (b <= -900.0)
		tmp = t_1;
	elseif (b <= 2.75e+14)
		tmp = Float64(Float64(x * (a ^ Float64(t + -1.0))) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (y * exp(b));
	tmp = 0.0;
	if (b <= -900.0)
		tmp = t_1;
	elseif (b <= 2.75e+14)
		tmp = (x * (a ^ (t + -1.0))) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -900.0], t$95$1, If[LessEqual[b, 2.75e+14], N[(N[(x * N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -900 or 2.75e14 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 91.9

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 91.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 84.7

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 84.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{0 - b}}{y}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{e^{0 - b}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      5. exp-diff N/A

         \[\leadsto \frac{x}{\frac{y}{\color{blue}{\frac{e^{0}}{e^{b}}}}} \]

      6. 1-exp N/A

         \[\leadsto \frac{x}{\frac{y}{\frac{\color{blue}{1}}{e^{b}}}} \]

      7. associate-/r/ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{1} \cdot e^{b}}} \]

      8. /-rgt-identity N/A

         \[\leadsto \frac{x}{\color{blue}{y} \cdot e^{b}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      10. exp-lowering-exp.f64 84.7

          \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 84.7%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   if -900 < b < 2.75e14

   1. Initial program 98.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 73.1

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 73.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{\log a \cdot \left(t - 1\right)}}{y}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot e^{\log a \cdot \left(t - 1\right)}}}{y} \]

      3. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \color{blue}{{a}^{\left(t - 1\right)}}}{y} \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{{a}^{\left(t - 1\right)}}}{y} \]

      5. sub-neg N/A

         \[\leadsto \frac{x \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}}{y} \]

      6. metadata-eval N/A

         \[\leadsto \frac{x \cdot {a}^{\left(t + \color{blue}{-1}\right)}}{y} \]

      7. +-lowering-+.f64 72.9

         \[\leadsto \frac{x \cdot {a}^{\color{blue}{\left(t + -1\right)}}}{y} \]

   8. Simplified 72.9%

      \[\leadsto \color{blue}{\frac{x \cdot {a}^{\left(t + -1\right)}}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 72.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y \cdot e^{b}}\\ \mathbf{if}\;b \leq -3500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 55000000000000:\\ \;\;\;\;{a}^{\left(t + -1\right)} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* y (exp b)))))
   (if (<= b -3500000.0)
     t_1
     (if (<= b 55000000000000.0) (* (pow a (+ t -1.0)) (/ x y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * exp(b));
	double tmp;
	if (b <= -3500000.0) {
		tmp = t_1;
	} else if (b <= 55000000000000.0) {
		tmp = pow(a, (t + -1.0)) * (x / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (y * exp(b))
    if (b <= (-3500000.0d0)) then
        tmp = t_1
    else if (b <= 55000000000000.0d0) then
        tmp = (a ** (t + (-1.0d0))) * (x / y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * Math.exp(b));
	double tmp;
	if (b <= -3500000.0) {
		tmp = t_1;
	} else if (b <= 55000000000000.0) {
		tmp = Math.pow(a, (t + -1.0)) * (x / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (y * math.exp(b))
	tmp = 0
	if b <= -3500000.0:
		tmp = t_1
	elif b <= 55000000000000.0:
		tmp = math.pow(a, (t + -1.0)) * (x / y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(y * exp(b)))
	tmp = 0.0
	if (b <= -3500000.0)
		tmp = t_1;
	elseif (b <= 55000000000000.0)
		tmp = Float64((a ^ Float64(t + -1.0)) * Float64(x / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (y * exp(b));
	tmp = 0.0;
	if (b <= -3500000.0)
		tmp = t_1;
	elseif (b <= 55000000000000.0)
		tmp = (a ^ (t + -1.0)) * (x / y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3500000.0], t$95$1, If[LessEqual[b, 55000000000000.0], N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.5e6 or 5.5e13 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 91.9

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 91.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 84.7

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 84.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{0 - b}}{y}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{e^{0 - b}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      5. exp-diff N/A

         \[\leadsto \frac{x}{\frac{y}{\color{blue}{\frac{e^{0}}{e^{b}}}}} \]

      6. 1-exp N/A

         \[\leadsto \frac{x}{\frac{y}{\frac{\color{blue}{1}}{e^{b}}}} \]

      7. associate-/r/ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{1} \cdot e^{b}}} \]

      8. /-rgt-identity N/A

         \[\leadsto \frac{x}{\color{blue}{y} \cdot e^{b}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      10. exp-lowering-exp.f64 84.7

          \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 84.7%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   if -3.5e6 < b < 5.5e13

   1. Initial program 98.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y} + 0} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} + 0 \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}, 0\right)} \]

   5. Simplified 83.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, {a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{y}, 0\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{y}}, 0\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{y}}, 0\right) \]

      2. exp-to-pow N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{{a}^{\left(t - 1\right)}}}{y}, 0\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{{a}^{\left(t - 1\right)}}}{y}, 0\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}}{y}, 0\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\left(t + \color{blue}{-1}\right)}}{y}, 0\right) \]

      6. +-lowering-+.f64 71.8

         \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\color{blue}{\left(t + -1\right)}}}{y}, 0\right) \]

   8. Simplified 71.8%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{{a}^{\left(t + -1\right)}}{y}}, 0\right) \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \frac{{a}^{\left(t + -1\right)}}{y}} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot {a}^{\left(t + -1\right)}}{y}} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot {a}^{\left(t + -1\right)}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{a}^{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      7. +-lowering-+.f64 N/A

         \[\leadsto {a}^{\color{blue}{\left(t + -1\right)}} \cdot \frac{x}{y} \]

      8. /-lowering-/.f64 64.8

         \[\leadsto {a}^{\left(t + -1\right)} \cdot \color{blue}{\frac{x}{y}} \]

   10. Applied egg-rr 64.8%

       \[\leadsto \color{blue}{{a}^{\left(t + -1\right)} \cdot \frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 45.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)\\ t_2 := 1 - t\_1\\ t_3 := \mathsf{fma}\left(b, b, b\right) \cdot \mathsf{fma}\left(b, b, b\right)\\ \mathbf{if}\;b \leq -3.05 \cdot 10^{+46}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-272}:\\ \;\;\;\;\frac{x \cdot \frac{a - a \cdot b}{a \cdot a}}{y}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{t\_2}{1 + \mathsf{fma}\left(b, b, b\right) \cdot t\_3} \cdot \left(1 + \mathsf{fma}\left(b, b, b\right) \cdot \left(-1 + \mathsf{fma}\left(b, b, b\right)\right)\right)}{a}}{y}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{t\_2}{1 - t\_3} \cdot \left(1 - \mathsf{fma}\left(b, b, b\right)\right)}{a}}{y}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1 - \mathsf{fma}\left(b, b, 0\right)}{1 + t\_1} \cdot \left(1 + b \cdot \left(b + -1\right)\right)}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{x}{y} \cdot \mathsf{fma}\left(0.5, b, -1\right), \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma b (fma b b 0.0) 0.0))
        (t_2 (- 1.0 t_1))
        (t_3 (* (fma b b b) (fma b b b))))
   (if (<= b -3.05e+46)
     (/ (* x (fma b (fma b (fma b -0.16666666666666666 0.5) -1.0) 1.0)) y)
     (if (<= b 2.55e-272)
       (/ (* x (/ (- a (* a b)) (* a a))) y)
       (if (<= b 1.15e+77)
         (/
          (*
           x
           (/
            (*
             (/ t_2 (+ 1.0 (* (fma b b b) t_3)))
             (+ 1.0 (* (fma b b b) (+ -1.0 (fma b b b)))))
            a))
          y)
         (if (<= b 5.6e+102)
           (/ (* x (/ (* (/ t_2 (- 1.0 t_3)) (- 1.0 (fma b b b))) a)) y)
           (if (<= b 9.2e+151)
             (/
              (*
               x
               (/
                (*
                 (/ (- 1.0 (fma b b 0.0)) (+ 1.0 t_1))
                 (+ 1.0 (* b (+ b -1.0))))
                a))
              y)
             (fma b (* (/ x y) (fma 0.5 b -1.0)) (/ x y)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, fma(b, b, 0.0), 0.0);
	double t_2 = 1.0 - t_1;
	double t_3 = fma(b, b, b) * fma(b, b, b);
	double tmp;
	if (b <= -3.05e+46) {
		tmp = (x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y;
	} else if (b <= 2.55e-272) {
		tmp = (x * ((a - (a * b)) / (a * a))) / y;
	} else if (b <= 1.15e+77) {
		tmp = (x * (((t_2 / (1.0 + (fma(b, b, b) * t_3))) * (1.0 + (fma(b, b, b) * (-1.0 + fma(b, b, b))))) / a)) / y;
	} else if (b <= 5.6e+102) {
		tmp = (x * (((t_2 / (1.0 - t_3)) * (1.0 - fma(b, b, b))) / a)) / y;
	} else if (b <= 9.2e+151) {
		tmp = (x * ((((1.0 - fma(b, b, 0.0)) / (1.0 + t_1)) * (1.0 + (b * (b + -1.0)))) / a)) / y;
	} else {
		tmp = fma(b, ((x / y) * fma(0.5, b, -1.0)), (x / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(b, fma(b, b, 0.0), 0.0)
	t_2 = Float64(1.0 - t_1)
	t_3 = Float64(fma(b, b, b) * fma(b, b, b))
	tmp = 0.0
	if (b <= -3.05e+46)
		tmp = Float64(Float64(x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y);
	elseif (b <= 2.55e-272)
		tmp = Float64(Float64(x * Float64(Float64(a - Float64(a * b)) / Float64(a * a))) / y);
	elseif (b <= 1.15e+77)
		tmp = Float64(Float64(x * Float64(Float64(Float64(t_2 / Float64(1.0 + Float64(fma(b, b, b) * t_3))) * Float64(1.0 + Float64(fma(b, b, b) * Float64(-1.0 + fma(b, b, b))))) / a)) / y);
	elseif (b <= 5.6e+102)
		tmp = Float64(Float64(x * Float64(Float64(Float64(t_2 / Float64(1.0 - t_3)) * Float64(1.0 - fma(b, b, b))) / a)) / y);
	elseif (b <= 9.2e+151)
		tmp = Float64(Float64(x * Float64(Float64(Float64(Float64(1.0 - fma(b, b, 0.0)) / Float64(1.0 + t_1)) * Float64(1.0 + Float64(b * Float64(b + -1.0)))) / a)) / y);
	else
		tmp = fma(b, Float64(Float64(x / y) * fma(0.5, b, -1.0)), Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(b * b + 0.0), $MachinePrecision] + 0.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b * b + b), $MachinePrecision] * N[(b * b + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.05e+46], N[(N[(x * N[(b * N[(b * N[(b * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 2.55e-272], N[(N[(x * N[(N[(a - N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 1.15e+77], N[(N[(x * N[(N[(N[(t$95$2 / N[(1.0 + N[(N[(b * b + b), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(b * b + b), $MachinePrecision] * N[(-1.0 + N[(b * b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 5.6e+102], N[(N[(x * N[(N[(N[(t$95$2 / N[(1.0 - t$95$3), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(b * b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 9.2e+151], N[(N[(x * N[(N[(N[(N[(1.0 - N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(b * N[(b + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(b * N[(N[(x / y), $MachinePrecision] * N[(0.5 * b + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if b < -3.04999999999999999e46

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 91.3

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 91.3%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 89.1

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 89.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right) + 1\right)}}{y} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1, 1\right)}}{y} \]

       3. sub-neg N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{y} \]

       4. metadata-eval N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \color{blue}{-1}, 1\right)}{y} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{-1}{6} \cdot b, -1\right)}, 1\right)}{y} \]

       6. +-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot b + \frac{1}{2}}, -1\right), 1\right)}{y} \]

       7. *-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{y} \]

       8. accelerator-lowering-fma.f64 84.8

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{y} \]

   11. Simplified 84.8%

       \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{y} \]

   if -3.04999999999999999e46 < b < 2.5499999999999999e-272

   1. Initial program 98.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 77.0

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 77.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 73.2

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 73.2%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 37.2

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 37.2%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. div-sub N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{a} - \frac{b}{a}\right)}}{y} \]

       2. frac-sub N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 \cdot a - a \cdot b}{a \cdot a}}}{y} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 \cdot a - a \cdot b}{a \cdot a}}}{y} \]

       4. *-lft-identity N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{a} - a \cdot b}{a \cdot a}}{y} \]

       5. --lowering--.f64 N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{a - a \cdot b}}{a \cdot a}}{y} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot \frac{a - \color{blue}{a \cdot b}}{a \cdot a}}{y} \]

       7. *-lowering-*.f64 42.4

          \[\leadsto \frac{x \cdot \frac{a - a \cdot b}{\color{blue}{a \cdot a}}}{y} \]

   13. Applied egg-rr 42.4%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{a - a \cdot b}{a \cdot a}}}{y} \]

   if 2.5499999999999999e-272 < b < 1.14999999999999997e77

   1. Initial program 99.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 75.2

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 75.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 57.5

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 57.5%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 35.0

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 35.0%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. flip3-- N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{{1}^{3} - {b}^{3}}{1 \cdot 1 + \left(b \cdot b + 1 \cdot b\right)}}}{a}}{y} \]

       2. metadata-eval N/A

          \[\leadsto \frac{x \cdot \frac{\frac{{1}^{3} - {b}^{3}}{\color{blue}{1} + \left(b \cdot b + 1 \cdot b\right)}}{a}}{y} \]

       3. flip3-+ N/A

          \[\leadsto \frac{x \cdot \frac{\frac{{1}^{3} - {b}^{3}}{\color{blue}{\frac{{1}^{3} + {\left(b \cdot b + 1 \cdot b\right)}^{3}}{1 \cdot 1 + \left(\left(b \cdot b + 1 \cdot b\right) \cdot \left(b \cdot b + 1 \cdot b\right) - 1 \cdot \left(b \cdot b + 1 \cdot b\right)\right)}}}}{a}}{y} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{{1}^{3} - {b}^{3}}{{1}^{3} + {\left(b \cdot b + 1 \cdot b\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(b \cdot b + 1 \cdot b\right) \cdot \left(b \cdot b + 1 \cdot b\right) - 1 \cdot \left(b \cdot b + 1 \cdot b\right)\right)\right)}}{a}}{y} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{{1}^{3} - {b}^{3}}{{1}^{3} + {\left(b \cdot b + 1 \cdot b\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(b \cdot b + 1 \cdot b\right) \cdot \left(b \cdot b + 1 \cdot b\right) - 1 \cdot \left(b \cdot b + 1 \cdot b\right)\right)\right)}}{a}}{y} \]

   13. Applied egg-rr 42.2%

       \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 - \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)}{1 + \mathsf{fma}\left(b, b, b\right) \cdot \left(\mathsf{fma}\left(b, b, b\right) \cdot \mathsf{fma}\left(b, b, b\right)\right)} \cdot \left(1 + \mathsf{fma}\left(b, b, b\right) \cdot \left(\mathsf{fma}\left(b, b, b\right) - 1\right)\right)}}{a}}{y} \]

   if 1.14999999999999997e77 < b < 5.60000000000000037e102

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 80.3

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 80.3%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 20.9

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 20.9%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 1.8

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 1.8%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. flip3-- N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{{1}^{3} - {b}^{3}}{1 \cdot 1 + \left(b \cdot b + 1 \cdot b\right)}}}{a}}{y} \]

       2. metadata-eval N/A

          \[\leadsto \frac{x \cdot \frac{\frac{{1}^{3} - {b}^{3}}{\color{blue}{1} + \left(b \cdot b + 1 \cdot b\right)}}{a}}{y} \]

       3. flip-+ N/A

          \[\leadsto \frac{x \cdot \frac{\frac{{1}^{3} - {b}^{3}}{\color{blue}{\frac{1 \cdot 1 - \left(b \cdot b + 1 \cdot b\right) \cdot \left(b \cdot b + 1 \cdot b\right)}{1 - \left(b \cdot b + 1 \cdot b\right)}}}}{a}}{y} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{{1}^{3} - {b}^{3}}{1 \cdot 1 - \left(b \cdot b + 1 \cdot b\right) \cdot \left(b \cdot b + 1 \cdot b\right)} \cdot \left(1 - \left(b \cdot b + 1 \cdot b\right)\right)}}{a}}{y} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{{1}^{3} - {b}^{3}}{1 \cdot 1 - \left(b \cdot b + 1 \cdot b\right) \cdot \left(b \cdot b + 1 \cdot b\right)} \cdot \left(1 - \left(b \cdot b + 1 \cdot b\right)\right)}}{a}}{y} \]

   13. Applied egg-rr 50.8%

       \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 - \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)}{1 - \mathsf{fma}\left(b, b, b\right) \cdot \mathsf{fma}\left(b, b, b\right)} \cdot \left(1 - \mathsf{fma}\left(b, b, b\right)\right)}}{a}}{y} \]

   if 5.60000000000000037e102 < b < 9.2000000000000003e151

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 87.7

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 87.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 21.9

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 21.9%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 9.6

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 9.6%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. flip-- N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 \cdot 1 - b \cdot b}{1 + b}}}{a}}{y} \]

       2. flip3-+ N/A

          \[\leadsto \frac{x \cdot \frac{\frac{1 \cdot 1 - b \cdot b}{\color{blue}{\frac{{1}^{3} + {b}^{3}}{1 \cdot 1 + \left(b \cdot b - 1 \cdot b\right)}}}}{a}}{y} \]

       3. associate-/r/ N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 \cdot 1 - b \cdot b}{{1}^{3} + {b}^{3}} \cdot \left(1 \cdot 1 + \left(b \cdot b - 1 \cdot b\right)\right)}}{a}}{y} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 \cdot 1 - b \cdot b}{{1}^{3} + {b}^{3}} \cdot \left(1 \cdot 1 + \left(b \cdot b - 1 \cdot b\right)\right)}}{a}}{y} \]

   13. Applied egg-rr 100.0%

       \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 - \mathsf{fma}\left(b, b, 0\right)}{1 + \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)} \cdot \left(1 + b \cdot \left(b + -1\right)\right)}}{a}}{y} \]

   if 9.2000000000000003e151 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 97.0

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 97.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 82.1

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 82.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{2} \cdot \frac{b \cdot x}{y}\right) + \frac{x}{y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{b \cdot x}{y} + -1 \cdot \frac{x}{y}\right)} + \frac{x}{y} \]

       2. associate-/l* N/A

          \[\leadsto b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot \frac{x}{y}\right)} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       3. associate-*r* N/A

          \[\leadsto b \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot b\right) \cdot \frac{x}{y}} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       4. *-commutative N/A

          \[\leadsto b \cdot \left(\color{blue}{\left(b \cdot \frac{1}{2}\right)} \cdot \frac{x}{y} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       5. associate-*r* N/A

          \[\leadsto b \cdot \left(\color{blue}{b \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right) + -1 \cdot \frac{x}{y}, \frac{x}{y}\right)} \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(b \cdot \frac{1}{2}\right) \cdot \frac{x}{y}} + -1 \cdot \frac{x}{y}, \frac{x}{y}\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(\frac{1}{2} \cdot b\right)} \cdot \frac{x}{y} + -1 \cdot \frac{x}{y}, \frac{x}{y}\right) \]

       9. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{x}{y} \cdot \left(\frac{1}{2} \cdot b + -1\right)}, \frac{x}{y}\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \left(\frac{1}{2} \cdot b + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right), \frac{x}{y}\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \color{blue}{\left(\frac{1}{2} \cdot b - 1\right)}, \frac{x}{y}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{x}{y} \cdot \left(\frac{1}{2} \cdot b - 1\right)}, \frac{x}{y}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{x}{y}} \cdot \left(\frac{1}{2} \cdot b - 1\right), \frac{x}{y}\right) \]

       14. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \color{blue}{\left(\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)\right)}, \frac{x}{y}\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \left(\frac{1}{2} \cdot b + \color{blue}{-1}\right), \frac{x}{y}\right) \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, -1\right)}, \frac{x}{y}\right) \]

       17. /-lowering-/.f64 28.8

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \mathsf{fma}\left(0.5, b, -1\right), \color{blue}{\frac{x}{y}}\right) \]

   11. Simplified 28.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{x}{y} \cdot \mathsf{fma}\left(0.5, b, -1\right), \frac{x}{y}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.05 \cdot 10^{+46}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-272}:\\ \;\;\;\;\frac{x \cdot \frac{a - a \cdot b}{a \cdot a}}{y}\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1 - \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)}{1 + \mathsf{fma}\left(b, b, b\right) \cdot \left(\mathsf{fma}\left(b, b, b\right) \cdot \mathsf{fma}\left(b, b, b\right)\right)} \cdot \left(1 + \mathsf{fma}\left(b, b, b\right) \cdot \left(-1 + \mathsf{fma}\left(b, b, b\right)\right)\right)}{a}}{y}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1 - \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)}{1 - \mathsf{fma}\left(b, b, b\right) \cdot \mathsf{fma}\left(b, b, b\right)} \cdot \left(1 - \mathsf{fma}\left(b, b, b\right)\right)}{a}}{y}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1 - \mathsf{fma}\left(b, b, 0\right)}{1 + \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)} \cdot \left(1 + b \cdot \left(b + -1\right)\right)}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{x}{y} \cdot \mathsf{fma}\left(0.5, b, -1\right), \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y \cdot e^{b}}\\ \mathbf{if}\;b \leq -900:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+17}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (* y (exp b)))))
   (if (<= b -900.0) t_1 (if (<= b 3.2e+17) (* x (/ (pow z y) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * exp(b));
	double tmp;
	if (b <= -900.0) {
		tmp = t_1;
	} else if (b <= 3.2e+17) {
		tmp = x * (pow(z, y) / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (y * exp(b))
    if (b <= (-900.0d0)) then
        tmp = t_1
    else if (b <= 3.2d+17) then
        tmp = x * ((z ** y) / y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (y * Math.exp(b));
	double tmp;
	if (b <= -900.0) {
		tmp = t_1;
	} else if (b <= 3.2e+17) {
		tmp = x * (Math.pow(z, y) / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (y * math.exp(b))
	tmp = 0
	if b <= -900.0:
		tmp = t_1
	elif b <= 3.2e+17:
		tmp = x * (math.pow(z, y) / y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(y * exp(b)))
	tmp = 0.0
	if (b <= -900.0)
		tmp = t_1;
	elseif (b <= 3.2e+17)
		tmp = Float64(x * Float64((z ^ y) / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (y * exp(b));
	tmp = 0.0;
	if (b <= -900.0)
		tmp = t_1;
	elseif (b <= 3.2e+17)
		tmp = x * ((z ^ y) / y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -900.0], t$95$1, If[LessEqual[b, 3.2e+17], N[(x * N[(N[Power[z, y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -900 or 3.2e17 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 91.9

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 91.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 84.7

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 84.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{0 - b}}{y}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{y}{e^{0 - b}}}} \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{y}{e^{0 - b}}}} \]

      5. exp-diff N/A

         \[\leadsto \frac{x}{\frac{y}{\color{blue}{\frac{e^{0}}{e^{b}}}}} \]

      6. 1-exp N/A

         \[\leadsto \frac{x}{\frac{y}{\frac{\color{blue}{1}}{e^{b}}}} \]

      7. associate-/r/ N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{y}{1} \cdot e^{b}}} \]

      8. /-rgt-identity N/A

         \[\leadsto \frac{x}{\color{blue}{y} \cdot e^{b}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot e^{b}}} \]

      10. exp-lowering-exp.f64 84.7

          \[\leadsto \frac{x}{y \cdot \color{blue}{e^{b}}} \]

   10. Applied egg-rr 84.7%

       \[\leadsto \color{blue}{\frac{x}{y \cdot e^{b}}} \]

   if -900 < b < 3.2e17

   1. Initial program 98.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

      3. log-lowering-log.f64 54.2

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

   5. Simplified 54.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot {z}^{y}}}{y} \]

      3. pow-lowering-pow.f64 53.5

         \[\leadsto \frac{x \cdot \color{blue}{{z}^{y}}}{y} \]

   8. Simplified 53.5%

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{{z}^{y}}{y}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{z}^{y}}{y}} \cdot x \]

      5. pow-lowering-pow.f64 53.5

         \[\leadsto \frac{\color{blue}{{z}^{y}}}{y} \cdot x \]

   10. Applied egg-rr 53.5%

       \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -900:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+17}:\\ \;\;\;\;x \cdot \frac{{z}^{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{+62}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-276}:\\ \;\;\;\;\frac{x \cdot \frac{a - a \cdot b}{a \cdot a}}{y}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1 - t\_1}{1 - \mathsf{fma}\left(b, b, b\right) \cdot \mathsf{fma}\left(b, b, b\right)} \cdot \left(1 - \mathsf{fma}\left(b, b, b\right)\right)}{a}}{y}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1 - \mathsf{fma}\left(b, b, 0\right)}{1 + t\_1} \cdot \left(1 + b \cdot \left(b + -1\right)\right)}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{x}{y} \cdot \mathsf{fma}\left(0.5, b, -1\right), \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma b (fma b b 0.0) 0.0)))
   (if (<= b -1.5e+62)
     (/ (* x (fma b (fma b (fma b -0.16666666666666666 0.5) -1.0) 1.0)) y)
     (if (<= b 6.5e-276)
       (/ (* x (/ (- a (* a b)) (* a a))) y)
       (if (<= b 5.6e+102)
         (/
          (*
           x
           (/
            (*
             (/ (- 1.0 t_1) (- 1.0 (* (fma b b b) (fma b b b))))
             (- 1.0 (fma b b b)))
            a))
          y)
         (if (<= b 9.2e+151)
           (/
            (*
             x
             (/
              (*
               (/ (- 1.0 (fma b b 0.0)) (+ 1.0 t_1))
               (+ 1.0 (* b (+ b -1.0))))
              a))
            y)
           (fma b (* (/ x y) (fma 0.5 b -1.0)) (/ x y))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(b, fma(b, b, 0.0), 0.0);
	double tmp;
	if (b <= -1.5e+62) {
		tmp = (x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y;
	} else if (b <= 6.5e-276) {
		tmp = (x * ((a - (a * b)) / (a * a))) / y;
	} else if (b <= 5.6e+102) {
		tmp = (x * ((((1.0 - t_1) / (1.0 - (fma(b, b, b) * fma(b, b, b)))) * (1.0 - fma(b, b, b))) / a)) / y;
	} else if (b <= 9.2e+151) {
		tmp = (x * ((((1.0 - fma(b, b, 0.0)) / (1.0 + t_1)) * (1.0 + (b * (b + -1.0)))) / a)) / y;
	} else {
		tmp = fma(b, ((x / y) * fma(0.5, b, -1.0)), (x / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(b, fma(b, b, 0.0), 0.0)
	tmp = 0.0
	if (b <= -1.5e+62)
		tmp = Float64(Float64(x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y);
	elseif (b <= 6.5e-276)
		tmp = Float64(Float64(x * Float64(Float64(a - Float64(a * b)) / Float64(a * a))) / y);
	elseif (b <= 5.6e+102)
		tmp = Float64(Float64(x * Float64(Float64(Float64(Float64(1.0 - t_1) / Float64(1.0 - Float64(fma(b, b, b) * fma(b, b, b)))) * Float64(1.0 - fma(b, b, b))) / a)) / y);
	elseif (b <= 9.2e+151)
		tmp = Float64(Float64(x * Float64(Float64(Float64(Float64(1.0 - fma(b, b, 0.0)) / Float64(1.0 + t_1)) * Float64(1.0 + Float64(b * Float64(b + -1.0)))) / a)) / y);
	else
		tmp = fma(b, Float64(Float64(x / y) * fma(0.5, b, -1.0)), Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(b * b + 0.0), $MachinePrecision] + 0.0), $MachinePrecision]}, If[LessEqual[b, -1.5e+62], N[(N[(x * N[(b * N[(b * N[(b * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 6.5e-276], N[(N[(x * N[(N[(a - N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 5.6e+102], N[(N[(x * N[(N[(N[(N[(1.0 - t$95$1), $MachinePrecision] / N[(1.0 - N[(N[(b * b + b), $MachinePrecision] * N[(b * b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(b * b + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 9.2e+151], N[(N[(x * N[(N[(N[(N[(1.0 - N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(b * N[(b + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(b * N[(N[(x / y), $MachinePrecision] * N[(0.5 * b + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.5e62

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 91.3

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 91.3%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 89.1

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 89.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right) + 1\right)}}{y} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1, 1\right)}}{y} \]

       3. sub-neg N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{y} \]

       4. metadata-eval N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \color{blue}{-1}, 1\right)}{y} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{-1}{6} \cdot b, -1\right)}, 1\right)}{y} \]

       6. +-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot b + \frac{1}{2}}, -1\right), 1\right)}{y} \]

       7. *-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{y} \]

       8. accelerator-lowering-fma.f64 84.8

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{y} \]

   11. Simplified 84.8%

       \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{y} \]

   if -1.5e62 < b < 6.49999999999999981e-276

   1. Initial program 98.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 77.0

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 77.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 73.2

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 73.2%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 37.2

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 37.2%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. div-sub N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{a} - \frac{b}{a}\right)}}{y} \]

       2. frac-sub N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 \cdot a - a \cdot b}{a \cdot a}}}{y} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 \cdot a - a \cdot b}{a \cdot a}}}{y} \]

       4. *-lft-identity N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{a} - a \cdot b}{a \cdot a}}{y} \]

       5. --lowering--.f64 N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{a - a \cdot b}}{a \cdot a}}{y} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot \frac{a - \color{blue}{a \cdot b}}{a \cdot a}}{y} \]

       7. *-lowering-*.f64 42.4

          \[\leadsto \frac{x \cdot \frac{a - a \cdot b}{\color{blue}{a \cdot a}}}{y} \]

   13. Applied egg-rr 42.4%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{a - a \cdot b}{a \cdot a}}}{y} \]

   if 6.49999999999999981e-276 < b < 5.60000000000000037e102

   1. Initial program 99.2%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 75.8

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 75.8%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 52.9

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 52.9%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 30.8

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 30.8%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. flip3-- N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{{1}^{3} - {b}^{3}}{1 \cdot 1 + \left(b \cdot b + 1 \cdot b\right)}}}{a}}{y} \]

       2. metadata-eval N/A

          \[\leadsto \frac{x \cdot \frac{\frac{{1}^{3} - {b}^{3}}{\color{blue}{1} + \left(b \cdot b + 1 \cdot b\right)}}{a}}{y} \]

       3. flip-+ N/A

          \[\leadsto \frac{x \cdot \frac{\frac{{1}^{3} - {b}^{3}}{\color{blue}{\frac{1 \cdot 1 - \left(b \cdot b + 1 \cdot b\right) \cdot \left(b \cdot b + 1 \cdot b\right)}{1 - \left(b \cdot b + 1 \cdot b\right)}}}}{a}}{y} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{{1}^{3} - {b}^{3}}{1 \cdot 1 - \left(b \cdot b + 1 \cdot b\right) \cdot \left(b \cdot b + 1 \cdot b\right)} \cdot \left(1 - \left(b \cdot b + 1 \cdot b\right)\right)}}{a}}{y} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{{1}^{3} - {b}^{3}}{1 \cdot 1 - \left(b \cdot b + 1 \cdot b\right) \cdot \left(b \cdot b + 1 \cdot b\right)} \cdot \left(1 - \left(b \cdot b + 1 \cdot b\right)\right)}}{a}}{y} \]

   13. Applied egg-rr 37.0%

       \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 - \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)}{1 - \mathsf{fma}\left(b, b, b\right) \cdot \mathsf{fma}\left(b, b, b\right)} \cdot \left(1 - \mathsf{fma}\left(b, b, b\right)\right)}}{a}}{y} \]

   if 5.60000000000000037e102 < b < 9.2000000000000003e151

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 87.7

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 87.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 21.9

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 21.9%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 9.6

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 9.6%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. flip-- N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 \cdot 1 - b \cdot b}{1 + b}}}{a}}{y} \]

       2. flip3-+ N/A

          \[\leadsto \frac{x \cdot \frac{\frac{1 \cdot 1 - b \cdot b}{\color{blue}{\frac{{1}^{3} + {b}^{3}}{1 \cdot 1 + \left(b \cdot b - 1 \cdot b\right)}}}}{a}}{y} \]

       3. associate-/r/ N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 \cdot 1 - b \cdot b}{{1}^{3} + {b}^{3}} \cdot \left(1 \cdot 1 + \left(b \cdot b - 1 \cdot b\right)\right)}}{a}}{y} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 \cdot 1 - b \cdot b}{{1}^{3} + {b}^{3}} \cdot \left(1 \cdot 1 + \left(b \cdot b - 1 \cdot b\right)\right)}}{a}}{y} \]

   13. Applied egg-rr 100.0%

       \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 - \mathsf{fma}\left(b, b, 0\right)}{1 + \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)} \cdot \left(1 + b \cdot \left(b + -1\right)\right)}}{a}}{y} \]

   if 9.2000000000000003e151 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 97.0

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 97.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 82.1

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 82.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{2} \cdot \frac{b \cdot x}{y}\right) + \frac{x}{y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{b \cdot x}{y} + -1 \cdot \frac{x}{y}\right)} + \frac{x}{y} \]

       2. associate-/l* N/A

          \[\leadsto b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot \frac{x}{y}\right)} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       3. associate-*r* N/A

          \[\leadsto b \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot b\right) \cdot \frac{x}{y}} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       4. *-commutative N/A

          \[\leadsto b \cdot \left(\color{blue}{\left(b \cdot \frac{1}{2}\right)} \cdot \frac{x}{y} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       5. associate-*r* N/A

          \[\leadsto b \cdot \left(\color{blue}{b \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right) + -1 \cdot \frac{x}{y}, \frac{x}{y}\right)} \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(b \cdot \frac{1}{2}\right) \cdot \frac{x}{y}} + -1 \cdot \frac{x}{y}, \frac{x}{y}\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(\frac{1}{2} \cdot b\right)} \cdot \frac{x}{y} + -1 \cdot \frac{x}{y}, \frac{x}{y}\right) \]

       9. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{x}{y} \cdot \left(\frac{1}{2} \cdot b + -1\right)}, \frac{x}{y}\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \left(\frac{1}{2} \cdot b + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right), \frac{x}{y}\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \color{blue}{\left(\frac{1}{2} \cdot b - 1\right)}, \frac{x}{y}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{x}{y} \cdot \left(\frac{1}{2} \cdot b - 1\right)}, \frac{x}{y}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{x}{y}} \cdot \left(\frac{1}{2} \cdot b - 1\right), \frac{x}{y}\right) \]

       14. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \color{blue}{\left(\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)\right)}, \frac{x}{y}\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \left(\frac{1}{2} \cdot b + \color{blue}{-1}\right), \frac{x}{y}\right) \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, -1\right)}, \frac{x}{y}\right) \]

       17. /-lowering-/.f64 28.8

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \mathsf{fma}\left(0.5, b, -1\right), \color{blue}{\frac{x}{y}}\right) \]

   11. Simplified 28.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{x}{y} \cdot \mathsf{fma}\left(0.5, b, -1\right), \frac{x}{y}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 17: 44.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{a} \cdot \left(1 + b\right)\\ \mathbf{if}\;b \leq -8.8 \cdot 10^{+54}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{x \cdot \left(\frac{1 - b}{a} \cdot t\_1\right)}{t\_1}}{y}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1 - \mathsf{fma}\left(b, b, 0\right)}{1 + \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)} \cdot \left(1 + b \cdot \left(b + -1\right)\right)}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{x}{y} \cdot \mathsf{fma}\left(0.5, b, -1\right), \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ 1.0 a) (+ 1.0 b))))
   (if (<= b -8.8e+54)
     (/ (* x (fma b (fma b (fma b -0.16666666666666666 0.5) -1.0) 1.0)) y)
     (if (<= b 1.35e+102)
       (/ (/ (* x (* (/ (- 1.0 b) a) t_1)) t_1) y)
       (if (<= b 9.2e+151)
         (/
          (*
           x
           (/
            (*
             (/ (- 1.0 (fma b b 0.0)) (+ 1.0 (fma b (fma b b 0.0) 0.0)))
             (+ 1.0 (* b (+ b -1.0))))
            a))
          y)
         (fma b (* (/ x y) (fma 0.5 b -1.0)) (/ x y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 / a) * (1.0 + b);
	double tmp;
	if (b <= -8.8e+54) {
		tmp = (x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y;
	} else if (b <= 1.35e+102) {
		tmp = ((x * (((1.0 - b) / a) * t_1)) / t_1) / y;
	} else if (b <= 9.2e+151) {
		tmp = (x * ((((1.0 - fma(b, b, 0.0)) / (1.0 + fma(b, fma(b, b, 0.0), 0.0))) * (1.0 + (b * (b + -1.0)))) / a)) / y;
	} else {
		tmp = fma(b, ((x / y) * fma(0.5, b, -1.0)), (x / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.0 / a) * Float64(1.0 + b))
	tmp = 0.0
	if (b <= -8.8e+54)
		tmp = Float64(Float64(x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y);
	elseif (b <= 1.35e+102)
		tmp = Float64(Float64(Float64(x * Float64(Float64(Float64(1.0 - b) / a) * t_1)) / t_1) / y);
	elseif (b <= 9.2e+151)
		tmp = Float64(Float64(x * Float64(Float64(Float64(Float64(1.0 - fma(b, b, 0.0)) / Float64(1.0 + fma(b, fma(b, b, 0.0), 0.0))) * Float64(1.0 + Float64(b * Float64(b + -1.0)))) / a)) / y);
	else
		tmp = fma(b, Float64(Float64(x / y) * fma(0.5, b, -1.0)), Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 / a), $MachinePrecision] * N[(1.0 + b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8.8e+54], N[(N[(x * N[(b * N[(b * N[(b * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 1.35e+102], N[(N[(N[(x * N[(N[(N[(1.0 - b), $MachinePrecision] / a), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 9.2e+151], N[(N[(x * N[(N[(N[(N[(1.0 - N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(b * N[(b * b + 0.0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(b * N[(b + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(b * N[(N[(x / y), $MachinePrecision] * N[(0.5 * b + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.7999999999999996e54

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 91.3

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 91.3%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 89.1

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 89.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right) + 1\right)}}{y} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1, 1\right)}}{y} \]

       3. sub-neg N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{y} \]

       4. metadata-eval N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \color{blue}{-1}, 1\right)}{y} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{-1}{6} \cdot b, -1\right)}, 1\right)}{y} \]

       6. +-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot b + \frac{1}{2}}, -1\right), 1\right)}{y} \]

       7. *-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{y} \]

       8. accelerator-lowering-fma.f64 84.8

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{y} \]

   11. Simplified 84.8%

       \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{y} \]

   if -8.7999999999999996e54 < b < 1.3500000000000001e102

   1. Initial program 98.6%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 76.5

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 76.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 63.7

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 63.7%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 34.3

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 34.3%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{1 - b}{a} \cdot x}}{y} \]

       2. div-sub N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{1}{a} - \frac{b}{a}\right)} \cdot x}{y} \]

       3. flip-- N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{a} \cdot \frac{1}{a} - \frac{b}{a} \cdot \frac{b}{a}}{\frac{1}{a} + \frac{b}{a}}} \cdot x}{y} \]

       4. associate-*l/ N/A

          \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{a} \cdot \frac{1}{a} - \frac{b}{a} \cdot \frac{b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}}{y} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{a} \cdot \frac{1}{a} - \frac{b}{a} \cdot \frac{b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}}{y} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(\frac{1}{a} \cdot \frac{1}{a} - \frac{b}{a} \cdot \frac{b}{a}\right) \cdot x}}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       7. difference-of-squares N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(\left(\frac{1}{a} + \frac{b}{a}\right) \cdot \left(\frac{1}{a} - \frac{b}{a}\right)\right)} \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       8. div-sub N/A

          \[\leadsto \frac{\frac{\left(\left(\frac{1}{a} + \frac{b}{a}\right) \cdot \color{blue}{\frac{1 - b}{a}}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(\left(\frac{1}{a} + \frac{b}{a}\right) \cdot \frac{1 - b}{a}\right)} \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       10. div-inv N/A

           \[\leadsto \frac{\frac{\left(\left(\color{blue}{1 \cdot \frac{1}{a}} + \frac{b}{a}\right) \cdot \frac{1 - b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       11. div-inv N/A

           \[\leadsto \frac{\frac{\left(\left(1 \cdot \frac{1}{a} + \color{blue}{b \cdot \frac{1}{a}}\right) \cdot \frac{1 - b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       12. distribute-rgt-out N/A

           \[\leadsto \frac{\frac{\left(\color{blue}{\left(\frac{1}{a} \cdot \left(1 + b\right)\right)} \cdot \frac{1 - b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{\frac{\left(\color{blue}{\left(\frac{1}{a} \cdot \left(1 + b\right)\right)} \cdot \frac{1 - b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       14. /-lowering-/.f64 N/A

           \[\leadsto \frac{\frac{\left(\left(\color{blue}{\frac{1}{a}} \cdot \left(1 + b\right)\right) \cdot \frac{1 - b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       15. +-lowering-+.f64 N/A

           \[\leadsto \frac{\frac{\left(\left(\frac{1}{a} \cdot \color{blue}{\left(1 + b\right)}\right) \cdot \frac{1 - b}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       16. /-lowering-/.f64 N/A

           \[\leadsto \frac{\frac{\left(\left(\frac{1}{a} \cdot \left(1 + b\right)\right) \cdot \color{blue}{\frac{1 - b}{a}}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

       17. --lowering--.f64 N/A

           \[\leadsto \frac{\frac{\left(\left(\frac{1}{a} \cdot \left(1 + b\right)\right) \cdot \frac{\color{blue}{1 - b}}{a}\right) \cdot x}{\frac{1}{a} + \frac{b}{a}}}{y} \]

   13. Applied egg-rr 37.3%

       \[\leadsto \frac{\color{blue}{\frac{\left(\left(\frac{1}{a} \cdot \left(1 + b\right)\right) \cdot \frac{1 - b}{a}\right) \cdot x}{\frac{1}{a} \cdot \left(1 + b\right)}}}{y} \]

   if 1.3500000000000001e102 < b < 9.2000000000000003e151

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 87.7

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 87.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 21.9

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 21.9%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 9.6

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 9.6%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. flip-- N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 \cdot 1 - b \cdot b}{1 + b}}}{a}}{y} \]

       2. flip3-+ N/A

          \[\leadsto \frac{x \cdot \frac{\frac{1 \cdot 1 - b \cdot b}{\color{blue}{\frac{{1}^{3} + {b}^{3}}{1 \cdot 1 + \left(b \cdot b - 1 \cdot b\right)}}}}{a}}{y} \]

       3. associate-/r/ N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 \cdot 1 - b \cdot b}{{1}^{3} + {b}^{3}} \cdot \left(1 \cdot 1 + \left(b \cdot b - 1 \cdot b\right)\right)}}{a}}{y} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 \cdot 1 - b \cdot b}{{1}^{3} + {b}^{3}} \cdot \left(1 \cdot 1 + \left(b \cdot b - 1 \cdot b\right)\right)}}{a}}{y} \]

   13. Applied egg-rr 100.0%

       \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 - \mathsf{fma}\left(b, b, 0\right)}{1 + \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)} \cdot \left(1 + b \cdot \left(b + -1\right)\right)}}{a}}{y} \]

   if 9.2000000000000003e151 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 97.0

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 97.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 82.1

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 82.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{2} \cdot \frac{b \cdot x}{y}\right) + \frac{x}{y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{b \cdot x}{y} + -1 \cdot \frac{x}{y}\right)} + \frac{x}{y} \]

       2. associate-/l* N/A

          \[\leadsto b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot \frac{x}{y}\right)} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       3. associate-*r* N/A

          \[\leadsto b \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot b\right) \cdot \frac{x}{y}} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       4. *-commutative N/A

          \[\leadsto b \cdot \left(\color{blue}{\left(b \cdot \frac{1}{2}\right)} \cdot \frac{x}{y} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       5. associate-*r* N/A

          \[\leadsto b \cdot \left(\color{blue}{b \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right) + -1 \cdot \frac{x}{y}, \frac{x}{y}\right)} \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(b \cdot \frac{1}{2}\right) \cdot \frac{x}{y}} + -1 \cdot \frac{x}{y}, \frac{x}{y}\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(\frac{1}{2} \cdot b\right)} \cdot \frac{x}{y} + -1 \cdot \frac{x}{y}, \frac{x}{y}\right) \]

       9. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{x}{y} \cdot \left(\frac{1}{2} \cdot b + -1\right)}, \frac{x}{y}\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \left(\frac{1}{2} \cdot b + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right), \frac{x}{y}\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \color{blue}{\left(\frac{1}{2} \cdot b - 1\right)}, \frac{x}{y}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{x}{y} \cdot \left(\frac{1}{2} \cdot b - 1\right)}, \frac{x}{y}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{x}{y}} \cdot \left(\frac{1}{2} \cdot b - 1\right), \frac{x}{y}\right) \]

       14. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \color{blue}{\left(\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)\right)}, \frac{x}{y}\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \left(\frac{1}{2} \cdot b + \color{blue}{-1}\right), \frac{x}{y}\right) \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, -1\right)}, \frac{x}{y}\right) \]

       17. /-lowering-/.f64 28.8

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \mathsf{fma}\left(0.5, b, -1\right), \color{blue}{\frac{x}{y}}\right) \]

   11. Simplified 28.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{x}{y} \cdot \mathsf{fma}\left(0.5, b, -1\right), \frac{x}{y}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.8 \cdot 10^{+54}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{x \cdot \left(\frac{1 - b}{a} \cdot \left(\frac{1}{a} \cdot \left(1 + b\right)\right)\right)}{\frac{1}{a} \cdot \left(1 + b\right)}}{y}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1 - \mathsf{fma}\left(b, b, 0\right)}{1 + \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)} \cdot \left(1 + b \cdot \left(b + -1\right)\right)}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{x}{y} \cdot \mathsf{fma}\left(0.5, b, -1\right), \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.1 \cdot 10^{+54}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;b \leq 5.7 \cdot 10^{-273}:\\ \;\;\;\;\frac{x \cdot \frac{a - a \cdot b}{a \cdot a}}{y}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{1 - \mathsf{fma}\left(b, b, 0\right)}{1 + \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)} \cdot \left(1 + b \cdot \left(b + -1\right)\right)}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{x}{y} \cdot \mathsf{fma}\left(0.5, b, -1\right), \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -5.1e+54)
   (/ (* x (fma b (fma b (fma b -0.16666666666666666 0.5) -1.0) 1.0)) y)
   (if (<= b 5.7e-273)
     (/ (* x (/ (- a (* a b)) (* a a))) y)
     (if (<= b 9.2e+151)
       (/
        (*
         x
         (/
          (*
           (/ (- 1.0 (fma b b 0.0)) (+ 1.0 (fma b (fma b b 0.0) 0.0)))
           (+ 1.0 (* b (+ b -1.0))))
          a))
        y)
       (fma b (* (/ x y) (fma 0.5 b -1.0)) (/ x y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -5.1e+54) {
		tmp = (x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y;
	} else if (b <= 5.7e-273) {
		tmp = (x * ((a - (a * b)) / (a * a))) / y;
	} else if (b <= 9.2e+151) {
		tmp = (x * ((((1.0 - fma(b, b, 0.0)) / (1.0 + fma(b, fma(b, b, 0.0), 0.0))) * (1.0 + (b * (b + -1.0)))) / a)) / y;
	} else {
		tmp = fma(b, ((x / y) * fma(0.5, b, -1.0)), (x / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -5.1e+54)
		tmp = Float64(Float64(x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y);
	elseif (b <= 5.7e-273)
		tmp = Float64(Float64(x * Float64(Float64(a - Float64(a * b)) / Float64(a * a))) / y);
	elseif (b <= 9.2e+151)
		tmp = Float64(Float64(x * Float64(Float64(Float64(Float64(1.0 - fma(b, b, 0.0)) / Float64(1.0 + fma(b, fma(b, b, 0.0), 0.0))) * Float64(1.0 + Float64(b * Float64(b + -1.0)))) / a)) / y);
	else
		tmp = fma(b, Float64(Float64(x / y) * fma(0.5, b, -1.0)), Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -5.1e+54], N[(N[(x * N[(b * N[(b * N[(b * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 5.7e-273], N[(N[(x * N[(N[(a - N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 9.2e+151], N[(N[(x * N[(N[(N[(N[(1.0 - N[(b * b + 0.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(b * N[(b * b + 0.0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(b * N[(b + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(b * N[(N[(x / y), $MachinePrecision] * N[(0.5 * b + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -5.10000000000000009e54

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 91.3

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 91.3%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 89.1

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 89.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right) + 1\right)}}{y} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1, 1\right)}}{y} \]

       3. sub-neg N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{y} \]

       4. metadata-eval N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \color{blue}{-1}, 1\right)}{y} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{-1}{6} \cdot b, -1\right)}, 1\right)}{y} \]

       6. +-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot b + \frac{1}{2}}, -1\right), 1\right)}{y} \]

       7. *-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{y} \]

       8. accelerator-lowering-fma.f64 84.8

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{y} \]

   11. Simplified 84.8%

       \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{y} \]

   if -5.10000000000000009e54 < b < 5.69999999999999972e-273

   1. Initial program 98.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 77.0

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 77.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 73.2

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 73.2%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 37.2

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 37.2%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. div-sub N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{a} - \frac{b}{a}\right)}}{y} \]

       2. frac-sub N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 \cdot a - a \cdot b}{a \cdot a}}}{y} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 \cdot a - a \cdot b}{a \cdot a}}}{y} \]

       4. *-lft-identity N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{a} - a \cdot b}{a \cdot a}}{y} \]

       5. --lowering--.f64 N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{a - a \cdot b}}{a \cdot a}}{y} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot \frac{a - \color{blue}{a \cdot b}}{a \cdot a}}{y} \]

       7. *-lowering-*.f64 42.4

          \[\leadsto \frac{x \cdot \frac{a - a \cdot b}{\color{blue}{a \cdot a}}}{y} \]

   13. Applied egg-rr 42.4%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{a - a \cdot b}{a \cdot a}}}{y} \]

   if 5.69999999999999972e-273 < b < 9.2000000000000003e151

   1. Initial program 99.3%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 76.9

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 76.9%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 50.0

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 50.0%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 28.9

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 28.9%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. flip-- N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 \cdot 1 - b \cdot b}{1 + b}}}{a}}{y} \]

       2. flip3-+ N/A

          \[\leadsto \frac{x \cdot \frac{\frac{1 \cdot 1 - b \cdot b}{\color{blue}{\frac{{1}^{3} + {b}^{3}}{1 \cdot 1 + \left(b \cdot b - 1 \cdot b\right)}}}}{a}}{y} \]

       3. associate-/r/ N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 \cdot 1 - b \cdot b}{{1}^{3} + {b}^{3}} \cdot \left(1 \cdot 1 + \left(b \cdot b - 1 \cdot b\right)\right)}}{a}}{y} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 \cdot 1 - b \cdot b}{{1}^{3} + {b}^{3}} \cdot \left(1 \cdot 1 + \left(b \cdot b - 1 \cdot b\right)\right)}}{a}}{y} \]

   13. Applied egg-rr 37.2%

       \[\leadsto \frac{x \cdot \frac{\color{blue}{\frac{1 - \mathsf{fma}\left(b, b, 0\right)}{1 + \mathsf{fma}\left(b, \mathsf{fma}\left(b, b, 0\right), 0\right)} \cdot \left(1 + b \cdot \left(b + -1\right)\right)}}{a}}{y} \]

   if 9.2000000000000003e151 < b

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 97.0

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 97.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 82.1

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 82.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{b \cdot \left(-1 \cdot \frac{x}{y} + \frac{1}{2} \cdot \frac{b \cdot x}{y}\right) + \frac{x}{y}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{b \cdot x}{y} + -1 \cdot \frac{x}{y}\right)} + \frac{x}{y} \]

       2. associate-/l* N/A

          \[\leadsto b \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(b \cdot \frac{x}{y}\right)} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       3. associate-*r* N/A

          \[\leadsto b \cdot \left(\color{blue}{\left(\frac{1}{2} \cdot b\right) \cdot \frac{x}{y}} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       4. *-commutative N/A

          \[\leadsto b \cdot \left(\color{blue}{\left(b \cdot \frac{1}{2}\right)} \cdot \frac{x}{y} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       5. associate-*r* N/A

          \[\leadsto b \cdot \left(\color{blue}{b \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right)} + -1 \cdot \frac{x}{y}\right) + \frac{x}{y} \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} \cdot \frac{x}{y}\right) + -1 \cdot \frac{x}{y}, \frac{x}{y}\right)} \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(b \cdot \frac{1}{2}\right) \cdot \frac{x}{y}} + -1 \cdot \frac{x}{y}, \frac{x}{y}\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(\frac{1}{2} \cdot b\right)} \cdot \frac{x}{y} + -1 \cdot \frac{x}{y}, \frac{x}{y}\right) \]

       9. distribute-rgt-out N/A

          \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{x}{y} \cdot \left(\frac{1}{2} \cdot b + -1\right)}, \frac{x}{y}\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \left(\frac{1}{2} \cdot b + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right), \frac{x}{y}\right) \]

       11. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \color{blue}{\left(\frac{1}{2} \cdot b - 1\right)}, \frac{x}{y}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{x}{y} \cdot \left(\frac{1}{2} \cdot b - 1\right)}, \frac{x}{y}\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{x}{y}} \cdot \left(\frac{1}{2} \cdot b - 1\right), \frac{x}{y}\right) \]

       14. sub-neg N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \color{blue}{\left(\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)\right)}, \frac{x}{y}\right) \]

       15. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \left(\frac{1}{2} \cdot b + \color{blue}{-1}\right), \frac{x}{y}\right) \]

       16. accelerator-lowering-fma.f64 N/A

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, -1\right)}, \frac{x}{y}\right) \]

       17. /-lowering-/.f64 28.8

           \[\leadsto \mathsf{fma}\left(b, \frac{x}{y} \cdot \mathsf{fma}\left(0.5, b, -1\right), \color{blue}{\frac{x}{y}}\right) \]

   11. Simplified 28.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{x}{y} \cdot \mathsf{fma}\left(0.5, b, -1\right), \frac{x}{y}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 19: 43.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{+54}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-275}:\\ \;\;\;\;\frac{x \cdot \frac{a - a \cdot b}{a \cdot a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -3.4e+54)
   (/ (* x (fma b (fma b (fma b -0.16666666666666666 0.5) -1.0) 1.0)) y)
   (if (<= b 3.5e-275) (/ (* x (/ (- a (* a b)) (* a a))) y) (/ (/ x a) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.4e+54) {
		tmp = (x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y;
	} else if (b <= 3.5e-275) {
		tmp = (x * ((a - (a * b)) / (a * a))) / y;
	} else {
		tmp = (x / a) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -3.4e+54)
		tmp = Float64(Float64(x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y);
	elseif (b <= 3.5e-275)
		tmp = Float64(Float64(x * Float64(Float64(a - Float64(a * b)) / Float64(a * a))) / y);
	else
		tmp = Float64(Float64(x / a) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -3.4e+54], N[(N[(x * N[(b * N[(b * N[(b * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[b, 3.5e-275], N[(N[(x * N[(N[(a - N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.4000000000000001e54

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 91.3

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 91.3%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 89.1

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 89.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right) + 1\right)}}{y} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1, 1\right)}}{y} \]

       3. sub-neg N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{y} \]

       4. metadata-eval N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \color{blue}{-1}, 1\right)}{y} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{-1}{6} \cdot b, -1\right)}, 1\right)}{y} \]

       6. +-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot b + \frac{1}{2}}, -1\right), 1\right)}{y} \]

       7. *-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{y} \]

       8. accelerator-lowering-fma.f64 84.8

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{y} \]

   11. Simplified 84.8%

       \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{y} \]

   if -3.4000000000000001e54 < b < 3.49999999999999969e-275

   1. Initial program 98.1%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 77.0

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 77.0%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 73.2

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 73.2%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 37.2

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 37.2%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. div-sub N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{a} - \frac{b}{a}\right)}}{y} \]

       2. frac-sub N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 \cdot a - a \cdot b}{a \cdot a}}}{y} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 \cdot a - a \cdot b}{a \cdot a}}}{y} \]

       4. *-lft-identity N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{a} - a \cdot b}{a \cdot a}}{y} \]

       5. --lowering--.f64 N/A

          \[\leadsto \frac{x \cdot \frac{\color{blue}{a - a \cdot b}}{a \cdot a}}{y} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot \frac{a - \color{blue}{a \cdot b}}{a \cdot a}}{y} \]

       7. *-lowering-*.f64 42.4

          \[\leadsto \frac{x \cdot \frac{a - a \cdot b}{\color{blue}{a \cdot a}}}{y} \]

   13. Applied egg-rr 42.4%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{a - a \cdot b}{a \cdot a}}}{y} \]

   if 3.49999999999999969e-275 < b

   1. Initial program 99.5%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 82.5

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 82.5%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 45.8

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 45.8%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 24.0

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 24.0%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

   12. Taylor expanded in b around 0

       \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 30.5

          \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]

   14. Simplified 30.5%

       \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 42.1% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{+27}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -2.4e+27)
   (/ (* x (fma b (fma b (fma b -0.16666666666666666 0.5) -1.0) 1.0)) y)
   (/ (/ x a) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -2.4e+27) {
		tmp = (x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y;
	} else {
		tmp = (x / a) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -2.4e+27)
		tmp = Float64(Float64(x * fma(b, fma(b, fma(b, -0.16666666666666666, 0.5), -1.0), 1.0)) / y);
	else
		tmp = Float64(Float64(x / a) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -2.4e+27], N[(N[(x * N[(b * N[(b * N[(b * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.39999999999999998e27

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 92.1

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 92.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 90.2

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 90.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1\right) + 1\right)}}{y} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) - 1, 1\right)}}{y} \]

       3. sub-neg N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right)}{y} \]

       4. metadata-eval N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, b \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot b\right) + \color{blue}{-1}, 1\right)}{y} \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{1}{2} + \frac{-1}{6} \cdot b, -1\right)}, 1\right)}{y} \]

       6. +-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot b + \frac{1}{2}}, -1\right), 1\right)}{y} \]

       7. *-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{b \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right)}{y} \]

       8. accelerator-lowering-fma.f64 78.6

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666, 0.5\right)}, -1\right), 1\right)}{y} \]

   11. Simplified 78.6%

       \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, -0.16666666666666666, 0.5\right), -1\right), 1\right)}}{y} \]

   if -2.39999999999999998e27 < b

   1. Initial program 98.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 79.6

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 79.6%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 57.5

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 57.5%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 29.9

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 29.9%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

   12. Taylor expanded in b around 0

       \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 33.6

          \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]

   14. Simplified 33.6%

       \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 40.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, b \cdot \mathsf{fma}\left(0.5, b, -1\right), x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -5.6e+26) (/ (fma x (* b (fma 0.5 b -1.0)) x) y) (/ (/ x a) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -5.6e+26) {
		tmp = fma(x, (b * fma(0.5, b, -1.0)), x) / y;
	} else {
		tmp = (x / a) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -5.6e+26)
		tmp = Float64(fma(x, Float64(b * fma(0.5, b, -1.0)), x) / y);
	else
		tmp = Float64(Float64(x / a) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -5.6e+26], N[(N[(x * N[(b * N[(0.5 * b + -1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.59999999999999999e26

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 92.1

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 92.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 90.2

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 90.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{x + b \cdot \left(-1 \cdot x + \frac{1}{2} \cdot \left(b \cdot x\right)\right)}}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\color{blue}{b \cdot \left(-1 \cdot x + \frac{1}{2} \cdot \left(b \cdot x\right)\right) + x}}{y} \]

       2. +-commutative N/A

          \[\leadsto \frac{b \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(b \cdot x\right) + -1 \cdot x\right)} + x}{y} \]

       3. distribute-lft-in N/A

          \[\leadsto \frac{\color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \left(b \cdot x\right)\right) + b \cdot \left(-1 \cdot x\right)\right)} + x}{y} \]

       4. associate-*r* N/A

          \[\leadsto \frac{\left(b \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot b\right) \cdot x\right)} + b \cdot \left(-1 \cdot x\right)\right) + x}{y} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\left(\color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot x} + b \cdot \left(-1 \cdot x\right)\right) + x}{y} \]

       6. mul-1-neg N/A

          \[\leadsto \frac{\left(\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot x + b \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) + x}{y} \]

       7. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\left(\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot x + \color{blue}{\left(\mathsf{neg}\left(b \cdot x\right)\right)}\right) + x}{y} \]

       8. distribute-lft-neg-in N/A

          \[\leadsto \frac{\left(\left(b \cdot \left(\frac{1}{2} \cdot b\right)\right) \cdot x + \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot x}\right) + x}{y} \]

       9. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(b \cdot \left(\frac{1}{2} \cdot b\right) + \left(\mathsf{neg}\left(b\right)\right)\right)} + x}{y} \]

       10. neg-mul-1 N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \left(\frac{1}{2} \cdot b\right) + \color{blue}{-1 \cdot b}\right) + x}{y} \]

       11. *-commutative N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \left(\frac{1}{2} \cdot b\right) + \color{blue}{b \cdot -1}\right) + x}{y} \]

       12. distribute-lft-in N/A

           \[\leadsto \frac{x \cdot \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot b + -1\right)\right)} + x}{y} \]

       13. metadata-eval N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \left(\frac{1}{2} \cdot b + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) + x}{y} \]

       14. sub-neg N/A

           \[\leadsto \frac{x \cdot \left(b \cdot \color{blue}{\left(\frac{1}{2} \cdot b - 1\right)}\right) + x}{y} \]

       15. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, b \cdot \left(\frac{1}{2} \cdot b - 1\right), x\right)}}{y} \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{b \cdot \left(\frac{1}{2} \cdot b - 1\right)}, x\right)}{y} \]

       17. sub-neg N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \color{blue}{\left(\frac{1}{2} \cdot b + \left(\mathsf{neg}\left(1\right)\right)\right)}, x\right)}{y} \]

       18. metadata-eval N/A

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \left(\frac{1}{2} \cdot b + \color{blue}{-1}\right), x\right)}{y} \]

       19. accelerator-lowering-fma.f64 65.2

           \[\leadsto \frac{\mathsf{fma}\left(x, b \cdot \color{blue}{\mathsf{fma}\left(0.5, b, -1\right)}, x\right)}{y} \]

   11. Simplified 65.2%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, b \cdot \mathsf{fma}\left(0.5, b, -1\right), x\right)}}{y} \]

   if -5.59999999999999999e26 < b

   1. Initial program 98.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 79.6

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 79.6%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 57.5

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 57.5%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 29.9

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 29.9%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

   12. Taylor expanded in b around 0

       \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 33.6

          \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]

   14. Simplified 33.6%

       \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 35.7% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+103}:\\ \;\;\;\;x \cdot \frac{1 - b}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -9e+103) (* x (/ (- 1.0 b) (* y a))) (/ (/ x a) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -9e+103) {
		tmp = x * ((1.0 - b) / (y * a));
	} else {
		tmp = (x / a) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-9d+103)) then
        tmp = x * ((1.0d0 - b) / (y * a))
    else
        tmp = (x / a) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -9e+103) {
		tmp = x * ((1.0 - b) / (y * a));
	} else {
		tmp = (x / a) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -9e+103:
		tmp = x * ((1.0 - b) / (y * a))
	else:
		tmp = (x / a) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -9e+103)
		tmp = Float64(x * Float64(Float64(1.0 - b) / Float64(y * a)));
	else
		tmp = Float64(Float64(x / a) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -9e+103)
		tmp = x * ((1.0 - b) / (y * a));
	else
		tmp = (x / a) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -9e+103], N[(x * N[(N[(1.0 - b), $MachinePrecision] / N[(y * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.00000000000000002e103

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 90.2

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 90.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 56.0

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 56.0%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 58.7

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 58.7%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   12. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{\frac{1 - b}{a}}{y}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\frac{1 - b}{a}}{y} \cdot x} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1 - b}{a}}{y} \cdot x} \]

       4. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{1 - b}{y \cdot a}} \cdot x \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 - b}{y \cdot a}} \cdot x \]

       6. --lowering--.f64 N/A

          \[\leadsto \frac{\color{blue}{1 - b}}{y \cdot a} \cdot x \]

       7. remove-double-div N/A

          \[\leadsto \frac{1 - b}{y \cdot \color{blue}{\frac{1}{\frac{1}{a}}}} \cdot x \]

       8. un-div-inv N/A

          \[\leadsto \frac{1 - b}{\color{blue}{\frac{y}{\frac{1}{a}}}} \cdot x \]

       9. remove-double-neg N/A

          \[\leadsto \frac{1 - b}{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)}}{\frac{1}{a}}} \cdot x \]

       10. neg-mul-1 N/A

           \[\leadsto \frac{1 - b}{\frac{\mathsf{neg}\left(\color{blue}{-1 \cdot y}\right)}{\frac{1}{a}}} \cdot x \]

       11. distribute-lft-neg-in N/A

           \[\leadsto \frac{1 - b}{\frac{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot y}}{\frac{1}{a}}} \cdot x \]

       12. metadata-eval N/A

           \[\leadsto \frac{1 - b}{\frac{\color{blue}{1} \cdot y}{\frac{1}{a}}} \cdot x \]

       13. associate-*l/ N/A

           \[\leadsto \frac{1 - b}{\color{blue}{\frac{1}{\frac{1}{a}} \cdot y}} \cdot x \]

       14. remove-double-div N/A

           \[\leadsto \frac{1 - b}{\color{blue}{a} \cdot y} \cdot x \]

       15. *-lowering-*.f64 56.6

           \[\leadsto \frac{1 - b}{\color{blue}{a \cdot y}} \cdot x \]

   13. Applied egg-rr 56.6%

       \[\leadsto \color{blue}{\frac{1 - b}{a \cdot y} \cdot x} \]

   if -9.00000000000000002e103 < b

   1. Initial program 98.9%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 80.6

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 80.6%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 57.7

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 57.7%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 30.0

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 30.0%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

   12. Taylor expanded in b around 0

       \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 33.0

          \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]

   14. Simplified 33.0%

       \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{+103}:\\ \;\;\;\;x \cdot \frac{1 - b}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 35.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \frac{1 - b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.1e+104) (* x (/ (- 1.0 b) y)) (/ (/ x a) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.1e+104) {
		tmp = x * ((1.0 - b) / y);
	} else {
		tmp = (x / a) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.1d+104)) then
        tmp = x * ((1.0d0 - b) / y)
    else
        tmp = (x / a) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.1e+104) {
		tmp = x * ((1.0 - b) / y);
	} else {
		tmp = (x / a) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -1.1e+104:
		tmp = x * ((1.0 - b) / y)
	else:
		tmp = (x / a) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.1e+104)
		tmp = Float64(x * Float64(Float64(1.0 - b) / y));
	else
		tmp = Float64(Float64(x / a) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -1.1e+104)
		tmp = x * ((1.0 - b) / y);
	else
		tmp = (x / a) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.1e+104], N[(x * N[(N[(1.0 - b), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / a), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.1e104

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 90.2

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 90.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 87.7

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 87.7%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{y} + \frac{x}{y}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{b \cdot x}{y}\right)\right)} + \frac{x}{y} \]

       2. associate-/l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{b \cdot \frac{x}{y}}\right)\right) + \frac{x}{y} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \frac{x}{y}} + \frac{x}{y} \]

       4. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \frac{x}{y}} \]

       5. +-commutative N/A

          \[\leadsto \color{blue}{\left(1 + \left(\mathsf{neg}\left(b\right)\right)\right)} \cdot \frac{x}{y} \]

       6. sub-neg N/A

          \[\leadsto \color{blue}{\left(1 - b\right)} \cdot \frac{x}{y} \]

       7. associate-/l* N/A

          \[\leadsto \color{blue}{\frac{\left(1 - b\right) \cdot x}{y}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(1 - b\right)}}{y} \]

       9. associate-*r/ N/A

          \[\leadsto \color{blue}{x \cdot \frac{1 - b}{y}} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \color{blue}{x \cdot \frac{1 - b}{y}} \]

       11. /-lowering-/.f64 N/A

           \[\leadsto x \cdot \color{blue}{\frac{1 - b}{y}} \]

       12. --lowering--.f64 46.7

           \[\leadsto x \cdot \frac{\color{blue}{1 - b}}{y} \]

   11. Simplified 46.7%

       \[\leadsto \color{blue}{x \cdot \frac{1 - b}{y}} \]

   if -1.1e104 < b

   1. Initial program 98.9%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 80.6

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 80.6%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(e^{\log a \cdot \left(t - 1\right)} + -1 \cdot \left(b \cdot e^{\log a \cdot \left(t - 1\right)}\right)\right)}}{y} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(-1 \cdot b\right) \cdot e^{\log a \cdot \left(t - 1\right)}}\right)}{y} \]

      2. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot \left(e^{\log a \cdot \left(t - 1\right)} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}}{y} \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right)} \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      7. --lowering--.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\color{blue}{\left(0 - b\right)} + 1\right) \cdot e^{\log a \cdot \left(t - 1\right)}\right)}{y} \]

      8. exp-to-pow N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot \color{blue}{{a}^{\left(t - 1\right)}}\right)}{y} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}\right)}{y} \]

      11. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + \color{blue}{-1}\right)}\right)}{y} \]

      12. +-lowering-+.f64 57.7

          \[\leadsto \frac{x \cdot \left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\color{blue}{\left(t + -1\right)}}\right)}{y} \]

   8. Simplified 57.7%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(\left(0 - b\right) + 1\right) \cdot {a}^{\left(t + -1\right)}\right)}}{y} \]

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

       2. --lowering--.f64 30.0

          \[\leadsto \frac{x \cdot \frac{\color{blue}{1 - b}}{a}}{y} \]

   11. Simplified 30.0%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{1 - b}{a}}}{y} \]

   12. Taylor expanded in b around 0

       \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 33.0

          \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]

   14. Simplified 33.0%

       \[\leadsto \frac{\color{blue}{\frac{x}{a}}}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 34.9% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+26}:\\ \;\;\;\;x \cdot \frac{1 - b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -1.3e+26) (* x (/ (- 1.0 b) y)) (/ x (* y a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.3e+26) {
		tmp = x * ((1.0 - b) / y);
	} else {
		tmp = x / (y * a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.3d+26)) then
        tmp = x * ((1.0d0 - b) / y)
    else
        tmp = x / (y * a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -1.3e+26) {
		tmp = x * ((1.0 - b) / y);
	} else {
		tmp = x / (y * a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -1.3e+26:
		tmp = x * ((1.0 - b) / y)
	else:
		tmp = x / (y * a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -1.3e+26)
		tmp = Float64(x * Float64(Float64(1.0 - b) / y));
	else
		tmp = Float64(x / Float64(y * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -1.3e+26)
		tmp = x * ((1.0 - b) / y);
	else
		tmp = x / (y * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -1.3e+26], N[(x * N[(N[(1.0 - b), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.30000000000000001e26

   1. Initial program 100.0%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot e^{\color{blue}{\log a \cdot \left(t - 1\right)} - b}}{y} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\left(\log a \cdot \left(t - 1\right) + 0\right)} - b}}{y} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t - 1, 0\right)} - b}}{y} \]

      3. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{\left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\color{blue}{\log \left(e^{\log a}\right)}, t - 1, 0\right) - b}}{y} \]

      5. rem-exp-log N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log \color{blue}{a}, t - 1, 0\right) - b}}{y} \]

      6. sub-neg N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + \left(\mathsf{neg}\left(1\right)\right)}, 0\right) - b}}{y} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, t + \color{blue}{-1}, 0\right) - b}}{y} \]

      8. +-lowering-+.f64 92.1

         \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(\log a, \color{blue}{t + -1}, 0\right) - b}}{y} \]

   5. Simplified 92.1%

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(\log a, t + -1, 0\right)} - b}}{y} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{x \cdot e^{\color{blue}{-1 \cdot b}}}{y} \]
   7. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{neg}\left(b\right)}}}{y} \]

      2. neg-sub0 N/A

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

      3. --lowering--.f64 90.2

         \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   8. Simplified 90.2%

      \[\leadsto \frac{x \cdot e^{\color{blue}{0 - b}}}{y} \]

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot x}{y} + \frac{x}{y}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{b \cdot x}{y}\right)\right)} + \frac{x}{y} \]

       2. associate-/l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{b \cdot \frac{x}{y}}\right)\right) + \frac{x}{y} \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \frac{x}{y}} + \frac{x}{y} \]

       4. distribute-lft1-in N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(b\right)\right) + 1\right) \cdot \frac{x}{y}} \]

       5. +-commutative N/A

          \[\leadsto \color{blue}{\left(1 + \left(\mathsf{neg}\left(b\right)\right)\right)} \cdot \frac{x}{y} \]

       6. sub-neg N/A

          \[\leadsto \color{blue}{\left(1 - b\right)} \cdot \frac{x}{y} \]

       7. associate-/l* N/A

          \[\leadsto \color{blue}{\frac{\left(1 - b\right) \cdot x}{y}} \]

       8. *-commutative N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(1 - b\right)}}{y} \]

       9. associate-*r/ N/A

          \[\leadsto \color{blue}{x \cdot \frac{1 - b}{y}} \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \color{blue}{x \cdot \frac{1 - b}{y}} \]

       11. /-lowering-/.f64 N/A

           \[\leadsto x \cdot \color{blue}{\frac{1 - b}{y}} \]

       12. --lowering--.f64 39.9

           \[\leadsto x \cdot \frac{\color{blue}{1 - b}}{y} \]

   11. Simplified 39.9%

       \[\leadsto \color{blue}{x \cdot \frac{1 - b}{y}} \]

   if -1.30000000000000001e26 < b

   1. Initial program 98.8%

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y} + 0} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} + 0 \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}, 0\right)} \]

   5. Simplified 72.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, {a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{y}, 0\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{y}}, 0\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{y}}, 0\right) \]

      2. exp-to-pow N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{{a}^{\left(t - 1\right)}}}{y}, 0\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{{a}^{\left(t - 1\right)}}}{y}, 0\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}}{y}, 0\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\left(t + \color{blue}{-1}\right)}}{y}, 0\right) \]

      6. +-lowering-+.f64 62.4

         \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\color{blue}{\left(t + -1\right)}}}{y}, 0\right) \]

   8. Simplified 62.4%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{{a}^{\left(t + -1\right)}}{y}}, 0\right) \]

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

       2. *-lowering-*.f64 29.8

          \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

   11. Simplified 29.8%

       \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+26}:\\ \;\;\;\;x \cdot \frac{1 - b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 32.0% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot a} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ x (* y a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / (y * a);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x / (y * a)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x / (y * a);
}

Python

def code(x, y, z, t, a, b):
	return x / (y * a)

Julia

function code(x, y, z, t, a, b)
	return Float64(x / Float64(y * a))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x / (y * a);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x / N[(y * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
2. Add Preprocessing

3. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} \]
4. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{\frac{x \cdot e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y} + 0} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}} + 0 \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{e^{y \cdot \log z + \log a \cdot \left(t - 1\right)}}{y}, 0\right)} \]

5. Simplified 69.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, {a}^{\left(t + -1\right)} \cdot \frac{{z}^{y}}{y}, 0\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{y}}, 0\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{e^{\log a \cdot \left(t - 1\right)}}{y}}, 0\right) \]

   2. exp-to-pow N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{{a}^{\left(t - 1\right)}}}{y}, 0\right) \]

   3. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{{a}^{\left(t - 1\right)}}}{y}, 0\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\color{blue}{\left(t + \left(\mathsf{neg}\left(1\right)\right)\right)}}}{y}, 0\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\left(t + \color{blue}{-1}\right)}}{y}, 0\right) \]

   6. +-lowering-+.f64 59.5

      \[\leadsto \mathsf{fma}\left(x, \frac{{a}^{\color{blue}{\left(t + -1\right)}}}{y}, 0\right) \]

8. Simplified 59.5%

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{{a}^{\left(t + -1\right)}}{y}}, 0\right) \]

9. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

    2. *-lowering-*.f64 29.1

       \[\leadsto \frac{x}{\color{blue}{a \cdot y}} \]

11. Simplified 29.1%

    \[\leadsto \color{blue}{\frac{x}{a \cdot y}} \]

12. Final simplification 29.1%

    \[\leadsto \frac{x}{y \cdot a} \]
13. Add Preprocessing

Alternative 26: 16.0% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (/ x y))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / y;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x / y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x / y;
}

Python

def code(x, y, z, t, a, b):
	return x / y

Julia

function code(x, y, z, t, a, b)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x / y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 99.1%

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \frac{x \cdot e^{\color{blue}{y \cdot \log z} - b}}{y} \]
4. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \frac{x \cdot e^{\color{blue}{\left(y \cdot \log z + 0\right)} - b}}{y} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

   3. log-lowering-log.f64 71.4

      \[\leadsto \frac{x \cdot e^{\mathsf{fma}\left(y, \color{blue}{\log z}, 0\right) - b}}{y} \]

5. Simplified 71.4%

   \[\leadsto \frac{x \cdot e^{\color{blue}{\mathsf{fma}\left(y, \log z, 0\right)} - b}}{y} \]

6. Taylor expanded in b around 0

   \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot {z}^{y}}}{y} \]

   3. pow-lowering-pow.f64 46.8

      \[\leadsto \frac{x \cdot \color{blue}{{z}^{y}}}{y} \]

8. Simplified 46.8%

   \[\leadsto \color{blue}{\frac{x \cdot {z}^{y}}{y}} \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{x}{y}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 16.9

       \[\leadsto \color{blue}{\frac{x}{y}} \]

11. Simplified 16.9%

    \[\leadsto \color{blue}{\frac{x}{y}} \]
12. Add Preprocessing

Developer Target 1: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {a}^{\left(t - 1\right)}\\ t_2 := \frac{x \cdot \frac{t\_1}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot t\_1}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (pow a (- t 1.0)))
        (t_2 (/ (* x (/ t_1 y)) (- (+ b 1.0) (* y (log z))))))
   (if (< t -0.8845848504127471)
     t_2
     (if (< t 852031.2288374073)
       (/ (* (/ x y) t_1) (exp (- b (* (log z) y))))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a ** (t - 1.0d0)
    t_2 = (x * (t_1 / y)) / ((b + 1.0d0) - (y * log(z)))
    if (t < (-0.8845848504127471d0)) then
        tmp = t_2
    else if (t < 852031.2288374073d0) then
        tmp = ((x / y) * t_1) / exp((b - (log(z) * y)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.pow(a, (t - 1.0));
	double t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * Math.log(z)));
	double tmp;
	if (t < -0.8845848504127471) {
		tmp = t_2;
	} else if (t < 852031.2288374073) {
		tmp = ((x / y) * t_1) / Math.exp((b - (Math.log(z) * y)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(a, (t - 1.0))
	t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * math.log(z)))
	tmp = 0
	if t < -0.8845848504127471:
		tmp = t_2
	elif t < 852031.2288374073:
		tmp = ((x / y) * t_1) / math.exp((b - (math.log(z) * y)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = a ^ Float64(t - 1.0)
	t_2 = Float64(Float64(x * Float64(t_1 / y)) / Float64(Float64(b + 1.0) - Float64(y * log(z))))
	tmp = 0.0
	if (t < -0.8845848504127471)
		tmp = t_2;
	elseif (t < 852031.2288374073)
		tmp = Float64(Float64(Float64(x / y) * t_1) / exp(Float64(b - Float64(log(z) * y))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a ^ (t - 1.0);
	t_2 = (x * (t_1 / y)) / ((b + 1.0) - (y * log(z)));
	tmp = 0.0;
	if (t < -0.8845848504127471)
		tmp = t_2;
	elseif (t < 852031.2288374073)
		tmp = ((x / y) * t_1) / exp((b - (log(z) * y)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Power[a, N[(t - 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] / N[(N[(b + 1.0), $MachinePrecision] - N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -0.8845848504127471], t$95$2, If[Less[t, 852031.2288374073], N[(N[(N[(x / y), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Exp[N[(b - N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -8845848504127471/10000000000000000) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 8520312288374073/10000000000) (/ (* (/ x y) (pow a (- t 1))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))